Anna Currey

Irina Illina

Dominique Fohr

DYNAMIC ADJUSTMENT OF LANGUAGE MODELS FOR AUTOMATIC SPEECH RECOGNITION USING WORD SIMILARITY

Keywords: ASR, language modeling, OOV, word embeddings, lexicon extension

Out-of-vocabulary (OOV) words can pose a particular problem for automatic speech recognition (ASR) of broadcast news. The language models (LMs) of ASR systems are typically trained on static corpora, whereas new words (particularly new proper nouns) are continually introduced in the media. Additionally, such OOVs are often content-rich proper nouns that are vital to understanding the topic. In this work, we explore methods for dynamically adding OOVs to language models by adapting the n-gram language model used in our ASR system. We propose two strategies: the first relies on finding in-vocabulary (IV) words similar to the OOVs, where word embeddings are used to define similarity. Our second strategy leverages a small contemporary corpus to estimate OOV probabilities. The models we propose yield improvements in perplexity over the baseline; in addition, the corpus-based approach leads to a significant decrease in proper noun error rate over the baseline in recognition experiments.

INTRODUCTION

Automatic speech recognition (ASR) systems are often trained on large but static text corpora and with a fixed vocabulary. For a system whose goal is to recognize speech about current events, this can pose a problem, since new words are continually introduced based on the events that occur. A particular issue is proper nouns (PNs): the names of newly important people or locations may not be in the vocabulary of the system, but recognizing them can be paramount to understanding the topic. It is not possible to simply find a large enough corpus to cover all of the important words, as novel names will always be introduced into a language. Therefore, a competent ASR system dealing with current events should accommodate adding new words to its vocabulary dynamically. Updating the n-gram language model (LM) of a deep neural network (DNN) ASR system1 by adding proper nouns to its vocabulary and estimating their parameters is the focus of this article.

Two main strategies for updating n-gram language models with domain-relevant out-of-vocabulary (OOV) words have been proposed in the literature: directly estimating parameters of n-grams containing the relevant OOVs, and classbased adaptation. Direct estimation approaches typically rely on a notion of similarity between OOV and in-vocabulary (IV) words to model OOV behavior. For example, Lecorvé, Gravier, and Sébillot [START_REF] Lecorvé | Automatically finding semantically consistent n-grams to add new words in LVCSR systems[END_REF] defined equivalence relations for words and for n-grams based on part-of-speech and semantic relatedness. Additionally, Qin [START_REF] Qin | Learning out-of-vocabulary words in automatic speech recognition[END_REF] suggested generating n-grams to add to a language model by taking an OOV's context and replacing the surrounding IVs with other similar IVs. In class-based adaptation, an OOV word is assigned to a word class and LM probabilities are estimated based on this classification [START_REF] Allauzen | Open vocabulary ASR for audiovisual document indexation[END_REF]. Extensions to the class-based adaptation model include using linguistic information to better inform classes [START_REF] Pražák | Language model adaptation using different class-based models[END_REF] and using word vectors to cluster OOV words into classes [START_REF] Naptali | Class-based n-gram language model for new words using out-of-vocabulary to in-vocabulary similarity[END_REF]. In addition, Martins, Teixeira, and Neto [START_REF] Martins | Automatic estimation of language model parameters for unseen words using morpho-syntactic contextual information[END_REF] experimented with updating a language model without any adaptation data. They achieved this by classifying unseen words using morpho-syntactic information.

In this work, we make a number of contributions to the task of dynamically adding OOV words to a language model without retraining the model. Our approaches are based on the idea of directly adding n-grams to the language model and estimating their parameters. First, we expand on the work done in [START_REF] Lecorvé | Automatically finding semantically consistent n-grams to add new words in LVCSR systems[END_REF] by using word embeddings to find similar IV words after which to model the behavior of the OOV. This allows us to use a similarity measure based on unlabeled text without prior knowledge. Second, we propose using a small corpus contemporary to the test data in order to inform estimates of the behavior of the OOV PNs. This method has the advantage of being easily applicable in a real-world setting, where textual news articles from a given time period could aid in recognition of news from that time period. In addition to being practical, this corpus-based method is also successful, yielding a statistically significant improvement in proper noun error rate over the baseline LM. Notably, the methods proposed in this paper are applied as part of a larger system in which many more OOV PNs are added to the language model than appear in the articles; these methods prove successful in handling this unique challenge.

METHODOLOGY

The overall process for recognizing OOV PNs in a news article using our system is as follows: first, the original ASR system is run to create a recognition hypothesis for the article. Based on that initial hypothesis, a list of OOV PNs that are likely to be in the article is generated [START_REF] Sheikh | Learning to retrieve out-ofvocabulary words in speech recognition[END_REF][START_REF] Sheikh | OOV proper name retrieval using topic and lexical context models[END_REF][START_REF] Sheikh | Document level semantic context for retrieving OOV proper names[END_REF]; the ranking of the OOV PNs is based on topic models using Latent Dirichlet Allocation. These new OOV PNs are added to the lexicon and to the language model, with the parameters of the language model being re-estimated to accommodate the new words. Finally, the system is re-run with the new language model.

The focus of this paper is dynamic re-estimation of the bigram LM used to produce the word lattice of the system. Our input is a baseline LM, a list of OOVs to be added for each article to be recognized, and a word-word similarity table. For some approaches, a corpus of textual data contemporary to the article is also used; we refer to this as the contemporary corpus.

From this input, our goal is to apply our estimation approaches in order to create a new LM that contains the vocabulary of the original LM, plus the added OOVs. This includes the following aspects:

• Unigram probability estimation for the OOVs (P (OOV ))

• Backoff weight estimation for the OOVs (B(OOV ))

• Finding bigrams containing OOVs (x-OOV and OOV-x)

• New bigram probability estimation (P (OOV |x) and

P (x|OOV ))
We propose two sets of approaches. Similarity-based approaches take advantage of word similarity measures to find similar IVs to a PN OOV; the behavior of the OOV is then modeled after the behavior of the similar IVs. Corpus-based approaches use the contemporary corpus to inform language model estimation. The behavior of the OOVs is based on their behavior in the contemporary corpus. Note that corpus-based approaches use word similarity as well, although the basis of these approaches is the contemporary corpus. Similaritybased approaches, on the other hand, do not use any outside corpus at language model estimation time (although outside data is used to create the pre-trained word similarity measure).

In all of our approaches, we use IV words that are similar to a given OOV to inform our LM estimates. In our experiments, our word similarity measures are based on word embeddings trained using the skip-gram architecture with a context window size of two words 2 proposed by Mikolov [START_REF] Mikolov | Efficient estimation of word representations in vector space[END_REF]. We make the assumption that the OOV PNs that we want to add are present in the corpus used for training the embeddings. For instance, a Wikipedia dump could be used in practice as it contains a huge number of (recent) PNs. Given the resulting vector representations of each word, we define word similarity between two words as the cosine similarity between their vector representations.

Similarity-based approaches

Similarity-based language model estimation approaches rely only on the behavior of similar words; no contemporary corpus is used. This is based on the hypothesis that an added OOV will behave more or less like the existing IVs that are most similar to it; here, we call such words similar IVs. Hence, we use the probabilities of n-grams containing those words to estimate the probabilities of n-grams containing the added OOV.

All of the proposed similarity-based approaches use the same overall process to estimate the language model parameters. For each added OOV: 5. Find all bigrams containing the similar IV; in some cases, select from among those bigrams 6. Add the new bigrams to the language model with the same probabilities, replacing the similar IV with the OOV Finally, once each OOV is added, the model is renormalized.

Unigram probability estimation

We study the following methods for estimating P (OOV ) based on similar IVs, where closestIV is the most similar IV word to a given added OOV:

• Closest IV: P (OOV ) ← P (closestIV )

• Maximum: P (OOV ) is the maximum probability of similar IVs

• Median: P (OOV ) is median probability of similar IVs 2 Small context window size is well-suited to modeling syntactic relations.

In our experiments, we use the five most similar IVs for the maximum and median estimation methods. Note that when the maximum method is used, a single similar IV gives its probability to the OOV; we refer to this IV as the IV used.

Estimation of backoff weights

Since the original language model uses modified Kneser-Ney smoothing, the original LM backoff weight for a given word depends on its bigrams. As a result, for estimating backoff weight for an OOV, we simply use the backoff weight of the IV used to find the OOV's bigrams (see below for how this IV is chosen). In addition, only the backoff weights of OOVs are changed; no existing backoff weights are modified.

Finding bigrams

We choose a single IV for each added OOV and model its bigrams after the bigrams of that IV. We study using the closest IV and the IV used to estimate unigram probability (when the unigram probability is defined using the maximum method).

In order to avoid adding too much noise to the adapted language model, we also study limiting the number of bigrams added for each OOV. For each OOV, we choose the M highest-probability bigrams to add to the language if M bigrams were not found for that OOV, we add all of the bigrams available.

Note that the bigrams added to the language model are kept separate from the bigrams in the original language model until all bigrams are found. So all new bigrams contain exactly one IV and one OOV. No bigrams containing two OOVs are added to the language model in the similarity-based approaches.

Bigram probability estimation

Each new bigram is based on a unique existing bigram, in which the OOV replaced an IV. Hence, the new bigram is assigned the probability of the corresponding existing bigram.

Corpus-based approaches

The data-driven word vectors that form the basis of our similarity-based approaches may be unreliable for the added OOV PNs, since these PNs were likely relatively rare in the corpus used to estimate the vectors. Our corpus-based approaches have the advantage of using both the word similarity measures and temporally relevant data (the contemporary corpus) to adapt the language model.

Corpus-based LM estimation is done as follows:

1. Calculate occurrence statistics of the OOV PNs in the contemporary corpus In our experiments, we adapt the baseline language model rather than the original language model3 ; an important consequence is that all of the added OOVs have an initial (very small) probability in the unadapted language model. Therefore, we propose using information about the behavior of the OOV PNs in the contemporary corpus to inflate their baseline probabilities (P base (OOV )). Specifically, we study two ways of defining unigram probabilities for the OOVs, where N (x) is the count of x in the contemporary corpus:

• Maximum likelihood:

P (OOV ) ← max P base (OOV ), N (OOV ) w N (w)
• Weighted by number of corpus occurrences of the OOV

Estimation of backoff weights

Backoff weights of the added OOVs (B(OOV )) are assigned based on the backoff weights of the closest IVs or of <unk>.

We do not attempt to train backoff weights using the corpus, as we expect overall behavior of the OOVs to be similar to that of the similar IVs or of <unk>, whereas training reliable Kneser-Ney backoff weights from the corpus would likely require much more information.

The methods for assigning backoff weights we study are:

• <unk>backoff: B(OOV ) ← B(<unk>) • Closest IV backoff: B(OOV ) ← B(closestIV )

Finding bigrams

One of the major advantages of having access to the contemporary corpus is that the corpus can be used to find bigrams. Our initial proposal for doing this is as follows: for each OOV, add a bigram containing that OOV if and only if that bigram was found in the corpus. These bigrams can be interpreted as safe choices: since they actually occur in the contemporary corpus, they may occur in the test data.

We study the following methods for using the corpus to find bigrams for a given added OOV:

• All corpus bigrams: add all bigrams from the corpus containing the OOV

• Limiting bigrams: only add bigrams that occur in the corpus more than a given amount of times

The limiting bigrams approach allows us to add only bigrams about which we are quite confident, while the all corpus bigrams approach allows us to increase bigram coverage. Note that when a bigram containing an added OOV PN and an OOV that is not added to the LM is found in the contemporary corpus, it is ignored, since we cannot add such bigrams to the language model. However, we do add bigrams consisting of two added OOV PNs to the LM; we call these OOV-OOV bigrams. Indeed, we do not expect this to be an edge case, since many of the added OOVs are names of people; the first and last names of a person often appear together.

Bigram probability estimation

We divide bigram probabilities into those of the form P (x|OOV ) (including OOV-OOV bigrams) and those of the form P (OOV |x). We consider two methods for estimating P (x|OOV ). The first consists of giving uniform probabilities to each bigram with the same first word. In the second method, we weight the bigram probability by the number of occurrences of the bigram in the contemporary corpus.

P (OOV |x) bigram probabilities can be estimated using information about the behavior of existing bigrams with the same first word, x. We refer to such existing bigrams as bigrams of the type x-y (where y is an IV), and to their probabilities as P (y|x). The following approaches to estimating P (OOV |x) are studied:

• Minimum: P (OOV |x) ← min y:P (y|x) =0 P (y|x)

• Closest IV: P (OOV |x) ← P (argmax y:P (y|x) =0 similarity(y, OOV )|x)

• Maximum of most similar IVs: P (OOV |x) ← max simIV s:P (simIV |x) =0 P (simIV |x), where simIV means similar IV

In the last case, we take the five most similar IVs such that P (simIV |x) = 0.

EXPERIMENTAL SETUP

3.1. Data

Training corpora

Training data was required to create the original language model, the word embeddings, and the lists of OOV PNs to add. For each of these tasks, some combination of the following three training sets was used:

• Le Monde + Gigaword: textual data from the French newspaper Le Monde and from the French Gigaword corpus (1B words; published between 1994 and 2008)

• Le Figaro: textual data from the French newspaper Le Figaro (8M words; 2014)

• L'Express: textual data from the French newspaper L'Express (51M words; 2014)

The original language model was trained using the Le Monde + Gigaword corpus. The word embeddings were created using a concatenation of the three corpora. The lists of OOV PNs to add were created using the L'Express corpus. The Le Figaro corpus was used as the contemporary corpus for corpus-based estimation, corresponding to the same time period as the development and test data.

Development corpus

The development corpus comes from the website of the news channel Euronews4 . It consists of 1,962 textual news articles (510,351 words) from January 2014 to June 2014. This corpus is used to evaluate each of our proposed approaches and select the approaches for which to run the test perplexity and recognition experiments. The OOV rate is 2.7% (13,768 words).

Test corpus

Once we have found the best parameter configurations and algorithms on the development corpus, we use the test corpus to further examine their performance. The test corpus consists of video reports from the Euronews website and their accompanying transcripts. The video reports are used for the recognition experiments, while the transcripts are used for the perplexity experiments. It is important to note that the reference transcriptions for the recognition experiments are the transcripts provided with the news videos, which may not always be an exact match to the audio. The articles were published in the first half of 2014. The test corpus consists of 467 articles (91,880 words), and the OOV rate is 2.3% (2,074 words).

Kaldi-based Automatic Transcription System

The Kaldi-based Automatic Transcription System (KATS) uses context dependent DNN-HMM phone models trained on 200-hour broadcast news audio files. The lexicon contains about 96K words. Using the SRILM toolkit [START_REF] Stolcke | SRILM-an extensible language modeling toolkit[END_REF], a bigram language model is estimated on the Le Monde + Gigaword corpus and used to produce the word lattice.

Language models

We use the original language model of the system, a baseline model, and an oracle model in our experiments. The original LM is a bigram model with modified Kneser-Ney smoothing trained on the Le Monde + Gigaword data.

Since we add words to the vocabularies of the adapted models, these models do not have the same vocabulary as the original LM. This means that we cannot compare the perplexities of these models with the perplexity of the original LM. As a result, we create the baseline language model, which has the same vocabulary as the adapted models. The baseline LM is created from the original LM by adding the words from the list of OOV PNs to be added. Only unigrams are added to the original LM to create the baseline. The probability mass assigned to the new unigrams is taken directly from <unk >, so renormalization is not required.

The oracle model is also created as a point of comparison to the adapted models. It represents the upper limit of how much the adapted models can be improved using the contemporary corpus. Like for the baseline model, the vocabulary of the oracle model consists of the vocabulary of the original language model plus the words in the list of OOVs to add. Our oracle model is an interpolation of the original LM and a language model estimated on the contemporary corpus.

Word lists

For each article, a ranked list of OOV PNs to be added to the LM was generated [START_REF] Sheikh | Document level semantic context for retrieving OOV proper names[END_REF]. We used the 128 words with the highest rankings to create the per-article added OOV lists. An automatic grapheme-to-phoneme conversion was used to assign pronunciations to the OOVs [START_REF] Illina | Grapheme-to-phoneme conversion using conditional random fields[END_REF].

The per-article added OOV lists are combined to form a list for the entire development corpus and one for the entire test corpus. The development OOV list contains 8,066 words, while the test OOV list contains 6,382 words.

RESULTS

We studied each of the proposed methods on the development corpus; the best-performing methods were then applied on the test corpus. Note that due to limitations in our original ASR system, we did not create per-article LMs 5 . Instead, we created a single LM for the whole development corpus and one for the whole test corpus for each method.

Results on the development corpus

For efficient experimentation, we divided adaptation into subtasks. For each subtask, we found the method that yielded the best result in terms of perplexity; we then used that method on all subsequent experiments. Table 1 shows the perplexities (PPL) on the development corpus when each of the unigram probability estimation methods was applied. Weighting the baseline probability by the number of corpus occurrences performed best overall, while taking the maximum probability of the similar IVs performed best among the similarity-based methods. However, we should note that there did not seem to be a large difference between the similarity-based methods. In subsequent experiments with similarity-based approaches, we estimated unigrams using maximum probability; for corpus-based approaches, we used weighting by corpus occurrences. Tables 2 and3 display the perplexity results on the development corpus using similarity-based and corpus-based bigram estimation, respectively. Note that, in the similaritybased approaches (table 2), we chose six and 24 bigrams to add per OOV because there were an average of six unique bigrams for each OOV in the contemporary corpus, while the oracle model added an average of 24 bigrams per added OOV.

Unigram

Bigram

Results on the test corpus

We selected the best-performing corpus-based and similaritybased methods from the experiments on the development data to be applied to the test data. For the similarity-based method, we used maximum probability for unigrams and added 24 bigrams per OOV. For the corpus-based method, we weighted unigram probabilities by corpus occurrences and assigned closest IV backoff. Bigrams were added with a cutoff of 5 occurrences in the contemporary corpus, with P (x|OOV ) uniform and P (OOV |x) assigned as the maximum probability of the most similar IVs. Since the similarity-based bigram approach did not outperform the similarity-based unigram-only approach on the development data, we included a similarity-based approach in which only unigrams were added to the LM. Table 4 displays the results of the perplexity experiments on the test data. These results are very similar to what was observed for the development data. All methods improved over the baseline; however, adding unigrams and bigrams using a similarity-based approach did not do as well as just adding unigrams. In addition, the corpus-based method performed better than either similarity-based method. The results for the recognition experiments on the test cor-pus are shown in table 5. We display the word error rate (WER) and proper noun error rate (PNER) on the test data for each of the adapted models, as well as for the original, baseline, and oracle language models. Statistical significance was calculated using the matched-pairs significance test described in [START_REF] Gillick | Some statistical issues in the comparison of speech recognition algorithms[END_REF]. These results closely mirror the development experiments and the perplexity results on the test data; namely, the corpus-based approaches outperform the similarity-based approaches and the baseline.

Method

Method

CONCLUSIONS

In this work, we set out to explore ways of improving the recognition of OOVs in an ASR system by adding OOVs to the language model. This was done by dynamically updating the LM without retraining it. We proposed two strategies: a similarity-based approach that took advantage of word embeddings to model behavior of added OOVs after that of IVs close to them in the vector space, and a corpus-based approach that used a small corpus from the same time period as the test data to find bigrams and estimate probabilities. The models we proposed yielded improvements in perplexity over the baseline; in addition, the corpus-based approach led to a significant decrease in PNER. This PNER decrease shows that new proper nouns were appropriately added to the language model. Unlike previous work on this topic, we found that using a small contemporary corpus was more successful in defining the LM behavior of the new OOVs than using the behavior of similar IVs.

1 .

 1 Find the similar IVs to the OOV 2. Estimate unigram P (OOV ) using unigram probabilities of similar IVs 3. Estimate B(OOV ) based on backoffs of similar IVs 4. Choose a similar IV to use for OOV bigram estimation

2 .

 2 Extract bigrams containing the added OOVs from the contemporary corpus 3. For each such OOV: (a) Find the similar IVs (b) Use the corpus statistics to estimate P (OOV ) (c) Estimate B(OOV ) based on similar IV behavior (d) Select contemporary corpus bigrams containing the OOV to add to the LM (e) Estimate bigram probabilities using information about similar IVs and corpus statistics 4. Renormalize the language model 2.2.1. Unigram probability estimation

Table 1 .

 1 Perplexity results for unigram probability estimates on the development corpus.

		estimation method	PPL
		baseline	230.4
		oracle	214.0
	similarity-closest IV probability	229.1
	based	maximum probability	228.2
		median probability	229.0
	corpus-	maximum likelihood	228.0
	based	weighted by corpus occur.	227.2

Table 2 .

 2 Perplexity results for similarity-based bigram estimation on the development corpus.From table 3, we can see that our best proposed corpusbased method yielded a large improvement over the baseline; the reduction in perplexity was 40.4% of the highest possible reduction (defined using the perplexity of the oracle model). In contrast to the similarity-based bigram methods in table 2, the corpus-based bigram methods resulted in an improvement over adding just unigrams to the LM (see table 1 for comparison).

		estimation method	PPL
		baseline	230.4
		oracle	214.0
		unigrams only	228.2
	finding	bigrams of closest IV	231.0
	bigrams	bigrams of IV used	232.3
	bigrams	all available bigrams added	231.0
	per OOV 6 bigrams added per OOV	228.3
		24 bigrams added per OOV	228.2

Table 3 .

 3 Perplexity results for corpus-based bigram estimation on the development corpus.

Table 4 .

 4 Perplexity results on the test data.

		PPL
	baseline	212.0
	oracle	197.4
	similarity-based unigrams	210.2
	similarity-based unigrams and bigrams 210.2
	corpus-based	206.6

Table 5 .

 5 Recognition results on the test data. Asterisks indicate statistically significant improvements over the baseline.

	WER PNER

State-of-the-art DNN ASR systems use an n-gram LM in combination with a recurrent neural network (RNN) LM.

The baseline LM is created from the original LM by giving a small probability to each added OOV. We adapt the baseline to ensure that all adapted LMs have the same vocabulary, allowing for direct perplexity comparisons.

http://fr.euronews.com/

Computing the Finite-State Transducer (FST) for each article to be recognized is too time consuming.

Acknowledgements

This work was funded by the ContNomina project supported by the French National Research Agency (ANR) under contract ANR-12-BS02-0009, and by an Amazon Academic Research Award.


Imre Bárány

CONVEX CONES, INTEGRAL ZONOTOPES, LIMIT SHAPE IMRE BÁRÁNY Julien Bureaux
email: julien.bureaux@math.cnrs.fr

Ben Lund
email: lund.ben@gmail.com

Convex cones

Keywords: integral zonotopes, limit shape 2010 Mathematics Subject Classification. Primary 52B20; Secondary 60C05, 05A17. Key words and phrases. convex cones, integer partitions, integer zonotopes, limit shape

come

Introduction and main results

This paper is about convex cones C in R d , integral zonotopes contained in C, and their limit shape. The cone C is going to be closed, convex and pointed (that is no line lies in C) and its interior, Int C, is non-empty. We write C or C d for the set of these cones.

A convex (lattice) polytope T ⊂ C is an integral zonotope if there exists m ∈ N and v 1 , . . . , v m ∈ Z d ∩ C (that is, each v i is lattice point in C) such that

T = m i=1 α i v i | (α 1 , . . . , α m ) ∈ [0, 1] m = Conv m i=1 ε i v i | (ε 1 , . . . , ε m ) ∈ {0, 1} m ,
The multiset V = {v 1 , . . . , v m } ⊂ Z d determines T = T (V ) uniquely, of course, but not conversely. More about this later. The endpoint of T is just m i=1 v i . Define T (C, k) as the family of all integral zonotopes in C whose endpoint is k ∈ Z d ∩ Int C. Clearly, T (C, k) is a finite set. Let p(C, k) denote its cardinality.

The main result of this paper is that, for large k, the overwhelming majority of the elements of T (C, k) are very close to a fixed convex set T 0 = T 0 (C, k) which is actually a zonoid. We write dist(A, B) for the Hausdorff distance of the sets A, B ⊂ R d . Here comes our main result. This result has been known for d = 2. Twenty years ago, Bárány [START_REF] Bárány | The limit shape of convex lattice polygons[END_REF], Sinai [START_REF] Ya | Probabilistic approach to the analysis of statistics for convex polygonal lines[END_REF] and Vershik [START_REF] Vershik | The limit shape of convex lattice polygons and related topics[END_REF] proved the existence of a limit shape for the set of all convex lattice polygons lying in the square [-n, n] 2 endowed with the uniform distribution. Although not all convex lattice polygons are zonotopes (or translates of zonotopes), Theorem 1.1 follows directly from their result. The approach of these papers relies on a natural link between convex lattice polygons on the first hand, and integer partitions on the other hand.

In addition to Theorem 1.1, the asymptotic behaviour as n → ∞ of p(C, nk) can also be determined.

Theorem 1.2. Under the above conditions on C and k there is a number q(C, k) > 0 such that, as n tends to infinity,

n -d d+1 log p(C, nk) -→ c d q(C, k),
where

c d = d+1 ζ(d+1) ζ(d) (d + 1
)! depends only on the dimension.

As we shall see in Section 3, q(C, k) is a constant multiple of the d + 1st root of the volume of the minimal cap of C containing k.

The next section connects integral zonotopes and strict integer partitions. Section 3 is about the limiting zonoid and some examples. The basic probabilistic model and proofs of the main results are in Section 4. Section 5 establishes several combinatorial properties of a typical zonotope T in T (C, k), namely we estimate the number of i-dimensional faces of T , for i = 0, 1, . . . , d -1. Section 6 includes proofs of the existence of limit shapes for integral zonotopes chosen in arbitrary convex bodies in R 2 , and in hypercubes for higher dimensions.

Strict integer partitions

A multiset V = {v 1 , . . . , v m } ⊂ Z d ∩ C determines the zonotope T = T (V ) uniquely but, as remarked earlier, T does not determine V uniquely. We are going to choose a suitable multiset W ⊂ Z d ∩ C uniquely. This is fairly simple. First let P d denote the primitive vectors in Z d ; a vector z = (z 1 , . . . , z d ) ∈ Z d is primitive if gcd(z 1 , . . . , z d ) = 1. Note that 0 / ∈ P d . Given T = T (V ) with generators V = {v 1 , . . . , v m } ⊂ Z d ∩ C, there is a unique multiset W = {w 1 , . . . , w } ⊂ P d ∩ C that generates the same zonotope, that is, T (V ) = T (W ). Indeed, each v i can be written uniquely as hw for some w ∈ P d and h ∈ Z + . Then put h copies of w in W . This way we get a multiset W ⊂ P d ∩ C such that T = T (W ). It is easy to check (we omit the details) that if T = T (U ) for some other multiset U ⊂ Z d ∩ C, then the above construction gives the same W . That means that W is uniquely determined by T .

We have just defined a one-to-one correspondence between lattice zonotopes T lying in C and multisets of primitive vectors

W ⊂ P d ∩ C. If the endpoint of T is k, then m i=1 w i = k.
Such a multiset is called a strict integer partition of the vector k ∈ Z d from the cone C. It can be alternatively described by the family (ω(x)) x∈P d of multiplicities ω(x) = card {j ∈ {1, . . . , } | w j = x} of the available parts x ∈ P d ∩ C. Notice that for any partition, there is only a finite number of vectors x ∈ P d ∩ C such that ω(x) = 0. Therefore, picking a strict partition is actually equivalent to picking a function ω : P d ∩ C → Z + with finite support. For any such function ω we define

X(ω) := x∈P d ∩C ω(x) x.
With this notation, the fact that ω describes a partition of k corresponds to the condition X(ω) = k.

One can consider non-strict partitions as well, that is, the uniform distribution on all multisets

V = {v 1 , . . . , v m } ⊂ Z d ∩ C with m 1 v i = k (where v i = 0)
, so the same zonotope may appear several times. The results of this paper remain valid in this case as well but some constants are different. For instance, Theorem 1.2 remains valid except that the constant c d is slightly different, namely, no division by ζ(d) is required. We omit the details.

The limiting zonoid

The dual C o of a cone C ∈ C d is defined, as usual, via

C o = {u ∈ R d | ∀x ∈ C \ {0}, u • x > 0}.
Note that the dual C o is an open cone, which is convenient for our purposes. Its closure is in C d as one can see easily. Given u ∈ C o and t > 0 we define the corresponding section C(u = t) and cap C(u ≤ t) of C by

C(u = t) = {x ∈ C | u • x = t}, C(u ≤ t) = {x ∈ C | u • x ≤ t}.
We are going to use the following result of Gigena [START_REF] Gigena | Integral invariants of convex cones[END_REF], see also [START_REF] Faraut | Analysis on symmetric cones, Oxford Mathematical Monographs[END_REF].

Theorem 3.1. Given C ∈ C d and a ∈ Int C there is a unique u = u(C, a) such that
• a is the center of gravity of the section C(u = 1),

• C(u ≤ 1) has minimal volume among all caps of C that contain a, moreover, C(u ≤ 1) is the unique cap with this property.

Clearly u ∈ C • . It follows that d d+1 a is the center of gravity of the cap C(u ≤ 1):

d d + 1 a = 1 Vol C(u ≤ 1) C(u≤1)
x dx.

Further, there is a λ > 0 such that a = C(u≤λ)

x dx.

One one can check directly that this λ is unique and is given by

λ = d + 1 d 1 Vol C(u ≤ 1) 1/(d+1) . Define Q = Q(C, a) = C(u ≤ λ) with this λ. An easy computation shows that q(C, a) := Vol Q(C, a) = λ d Vol C(u ≤ 1) = 1 + 1 d d Vol C(u ≤ 1) 1/(d+1)
.

We will show later in Section 4.5, that this is the q(C, k) appearing in Theorem 1.2.

h T 0 (v) = v • x dx
where the integral is taken over Q(C, k) ∩ C(v ≥ 0). Similarly, the point t(v) where the hyperplane orthogonal to v supports T 0 (C, k) is xdx with the integral taken over the same set as above. So the boundary point of T 0 with outer normal v is given by

(3.1) t(v) = Q(C,k)∩C(v≥0)
x dx. x dx scaled properly as explained in the Remark above. The computation is easy when d = 2. Then v = (t, -s) with s, t ≥ 0 and s+t = 1, say. Then ∩C(v ≥ 0) is a triangle with vertices (0, 0), (1, 0), (s, t). The integral in question is equal to the area of the triangle (which is t/2) times its centre of gravity (which equals (1 + s, t)/3). Using s = 1 -t and suitable scaling we get t(v) = (2t -t 2 , t 2 ). With x, y coordinates this is just x + y = 2 √ y (for x ≥ y), a parabola arc. Thus the boundary of T 0 consists of two parabola arcs given by the equations x + y = 2 √ y (for x ≤ y) and

As expected, t(v) = k for v ∈ C • and t(v) = 0 for v ∈ -C • .
x + y = 2 √ y (for x ≥ y), which is the same as the limit shape in R 2 of convex lattice polygons, see Bárány [START_REF] Bárány | The limit shape of convex lattice polygons[END_REF] and Vershik [START_REF] Vershik | The limit shape of convex lattice polygons and related topics[END_REF]. The same method works in R 3 . This time the previous triangle is replaced by the simplex with vertices (0, 0, 0), (1, 0, 0), (s, t, 0), (u, 0, v) with s, t, u, v ≥ 0 and s + t = 1 u + v = 1. The integral in question is the volume of this simplex (tv/6) times its centre of gravity ((1 + u + s, t, v)/4). Using s = 1 -t and u = 1 -v again we get t(v) = (3tv -t 2 v -tv 2 , t 2 v, tv 2 ) . Thus the equation of the boundary of T 0 is x + y + z = 3 3 √ tv; this holds when x -2y + z ≥ 0 and x + y -2z ≥ 0, as one can check directly. T 0 is centrally symmetric with respect to center (1/2, 1/2, 1/2). Its boundary is made up of six pieces that come in symmetric pairs. The piece in the region determined by inequalities x -2y + z ≥ 0 and x + y -2z ≥ 0 is given by the equation x + y + z = 3 3 √ yz. Other pieces are given by equations x + y + z = 3 3 √ xz and x + y + z = 3 3 √ xy, and the reflections with respect to the center.

The same method works in higher dimensions. There ∩ C(v ≥ 0) is not a single simplex, one has to triangulate it into simplices, and on each simplex, the integral in question can be computed the same way as above. We have not carried out this computation. Yet one can show that a certain part of the boundary of T 0 in R d is described by the equation The same limit shape comes up in another way as well. We are going to explain, rather informally, how this happens. We start with a simple proposition. Proposition 3.2. Assume S ⊂ C is a closed set and S x dx = a. Then Vol S ≤ q(C, a) and equality holds iff S = Q(C, k).

x 1 + . . . + x d = d(x 2 . . . x d ) 1/d .

Proof. It is clear that

S x dx = a implies that S u • x dx = u • a. Note that Q u • x dx = u • a as well. Then S\Q u • x dx = Q\S u • x dx.
The value of u • x is larger on S \ Q than on Q \ S. This shows that Vol S \ Q is smaller than Vol(Q \ S) unless both are equal to zero.

Assume that k ∈ Z d ∩ Int C and consider the cap Q(C, nk) when n is large. Define V n = Z d ∩ Q(C, nk).
It is known (and follows from an estimate similar to the one in Lemma A.2) that, as n goes to infinity, This shows that the (asymptotically) largest set S ⊂ Z d ∩C with x∈S x = nk is V n . The integral zonotope T (V n ) has its endpoint close to nk. The limiting zonoid T 0 (C, k) from Theorem 1.1 turns out to be the limit of the zonotopes 1 n T (V n ) as n → ∞.

The probabilistic approach

Throughout this paper we work under the following condition:

(4.1) C ∈ C d and k ∈ Int C ∩ Z d are fixed.
In this section, we give a proof of Theorem 1.1 and Theorem 1.2 based on statistical mechanics techniques. Obviously, the statement of the theorem itself is already of probabilistic nature.

Description of the model.

Let Ω be the set of all functions from P d ∩ C to Z + with finite support. Recall from Section 2 the correspondence between lattice zonotopes, strict integer partitions, and multiplicity functions ω ∈ Ω. In particular, each function ω is associated to a unique zonotope T (ω), and conversely. Moreover, the endpoint of T (ω) is

X(ω) = x∈P d ∩C ω(x) x.
The probability distribution Q n on Ω, which is defined for all ω ∈ Ω by

Q n (ω) = 1 p(C, nk) 1 {X(ω)=nk}
is exactly the uniform distribution on T (C, nk). The conclusion of Theorem 1.1 can be stated as convergence in probability for this distribution:

dist( 1 n T, T 0 ) Qn ---→ n→∞ 0.
We now define a new probability distribution P n on Ω, which will turn out to behave roughly like Q n when n tends to +∞. It depends on two parameters u and β n fixed throughout the paper as follows:

(4.2) u = u(C, λk), β n = d+1 ζ(d + 1) ζ(d) 1 n , where u(C, λk) = λ -1 u(C, k) ∈ C • is defined by Theorem 3.1, with λ > 0 chosen such that (d + 1)! C(u≤1)
x dx = k.

The reason for these choices will become apparent in Proposition 4.1. The probability distribution P n is then defined for all ω ∈ Ω by

P n (ω) = 1 Z n e -βnu•X(ω) , where Z n = ω∈Ω e -βnu•X(ω) .
This definition, which is directly inspired by statistical mechanics, is a special case of the Boltzmann distribution. It is a generalization of the model introduced by Sinai [START_REF] Ya | Probabilistic approach to the analysis of statistics for convex polygonal lines[END_REF] for convex lattice polygonal lines. Here Z n is the so-called partition function of the model. The sum defining Z n is easily seen to be convergent. The crucial observation is the following: since P n (ω) only depends on the value X(ω), we see that for all m ∈ Z d ,

(4.3) P n [X = m] = p(C, m) Z n e -βnu•m .
In particular, the distribution induced on Ω by P n conditional on the event

{X = nk} is exactly Q n . For all events E ⊂ Ω, Q n [E] = P n [E | X = nk] = P n [E ∩ {X = nk}] P n [X = nk] .
Consequently, our strategy of proof for Theorem 1.1 is to establish first a strong limit shape result for the distribution P n and then use this relation with E = {dist( 1 n T, T 0 ) > ε} to get the result for Q n . This deduction will only be possible if we show that P n [X = nk] is not too small.

In comparison with Q n , the distribution P n has a much simpler probabilistic structure. It is a product distribution on Ω since, by definition of

X, e -βnu•X(ω) = x∈P d ∩C e -βnω(x)u•x .
Thus, the family of random variables (ω(x)) x∈P d ∩C is independent under P n .

In addition, one can see that for all x ∈ P d ∩ C, the integer-valued random variable ω(x) has geometric distribution with parameter 1 -e -βnu•x , that is to say

(4.4) P n [ω(x) = i] = (1 -e -βnu•x ) e -iβnu•x , i ∈ Z + .
These two facts lead to a remarkable factorization of Z n as convergent product which is similar to Euler's formula,

Z n = x∈P d ∩C 1 1 -e -βnu•x .
Taking logarithms, we obtain finally a series expansion for log Z n , (4.5) log

Z n = x∈P d ∩C r≥1 e -rβnu•x r .
The same expression is used in [START_REF] Bárány | The limit shape of convex lattice polygons[END_REF] and [START_REF] Ya | Probabilistic approach to the analysis of statistics for convex polygonal lines[END_REF].

4.2. Asymptotic behaviour after rescaling. In this section, we investigate the asymptotic distribution of the random vector X. It turns out that, after a proper rescaling, it can described completely by the Laplace transform (also called a characteristic function in [START_REF] Faraut | Analysis on symmetric cones, Oxford Mathematical Monographs[END_REF]) of the cone C, which is defined by

Λ C (v) = C e -v•x dx, v ∈ C • .
When there is no risk of confusion with another cone, we write Λ = Λ C .

Proposition 4.1. Let µ n and Γ n denote respectively the mean value and the covariance matrix of the random vector X under the distribution P n . Then

(4.6) lim n→+∞ 1 n µ n = -∇Λ(u) = k, and lim n→+∞ n -d+2 d+1 Γ n = ∇ 2 Λ(u),
where ∇Λ and ∇ 2 Λ denote respectively the gradient and the Hessian matrix. Moreover, the distribution of X satisfies a central limit theorem in the sense that

Γ -1/2 n (X -µ n ) law -----→ n→+∞ N (0, I d ).
Proof. We introduce a function Z :

C • → R + such that Z n = Z(β n u). It is defined for all v ∈ C • by Z(v) = ω∈Ω e -v•X(ω) .
Let β = β n in the proof. We start with the statement about µ n . It is a well known fact that in such an exponential model, the first moments of X are given by the logarithmic derivatives of the function Z. In the case of the mean value, it results from the following simple computation:

µ n = E n [X] = ω∈Ω e -βu•X(ω) Z(βu) X(ω) = - ∇Z(βu) Z(βu) = -∇ log Z(βu).
Now, a straightforward generalization of formula (4.5) to the function Z leads to a series expansion of ∇ log Z, namely

-∇ log Z(βu) = r≥1 x∈P d ∩C e -rβu•x x.
For every index r less than 1/β, we approximate the summation on P d ∩ C with the corresponding d-dimensional integral by using Proposition A.1.

Since the terms with r > 1/β contribute only O(1/β), we obtain after summation

µ n = 1 β d+1 ζ(d + 1) ζ(d) C e -u•x x dx + O 1 β d in the limit β → 0.
The integral here is obviously -∇Λ(u). We need check that it is also equal to k. This is done by applying the fundamental theorem of calculus and the Fubini-Tonelli theorem:

C e -u•x x dx = C ∞ u•x e -t dt x dx = ∞ 0 e -t C(u≤t)
x dx dt.

An homothetic change of variable yields therefore by homogeneity:

C e -u•x x dx = ∞ 0 t d+1 e -t dt C(u≤1) x dx = (d + 1)! C(u≤1) x dx = k.
So the first part of the proposition is finally proven by replacing β with its expression as a function of n.

The proof of the asymptotic behaviour of the covariance matrix Γ n is entirely similar except that we consider the second order derivatives of the logarithmic partition function. We omit the details.

We now turn to the central limit result. Recall that the random variables ω(x) are independent and each ω(x) follows the geometric distribution with parameter e -βu•x . In particular, Γ

-1/2 n (X-µ n ) is the sum of the independent random variables (ω(x) -E[ω(x)]) Γ -1/2 n
x of mean value 0. We are going to check the classical Lyapunov condition on third moments, which implies the central limit theorem for independent, but not necessarily identically distributed, summands. We must show that

L n = x∈P d ∩C E (ω(x) -E[ω(x)]) Γ -1/2 n x 3
goes to 0 as n tends to +∞. Elementary computations on the geometric distribution yield, for all x ∈ P d ∩ C,

E[ω(x)] = e -βu•x 1 -e -βu•x , and E |ω(x) -E[ω(x)]| 3 ≤ 3 e -βu•x (1 -e -βu•x ) 3 . Letting Γ -1/2 n
denote the operator norm, we obtain therefore,

L n ≤ 3 Γ -1/2 n 3 x∈P d ∩C x 3 e -βu•x (1 -e -βu•x ) 3 .
From the first part of the proposition, we already know that Γ -1/2 n is of order n -1 2 (d+2)/(d+1) . Moreover, Proposition A.1 shows that

x∈P d ∩C x 3 e -βu•x (1 -e -βu•x ) 3 and n d+3 d+1 C
x 3 e -u•x dx are of the same order of magnitude as n tends to +∞. Accordingly, L n is at most of order n -d/(2d+2) , hence tends to 0 and we can indeed apply Lyapunov's central limit theorem. The proof is complete.

Letting σ n = n d+2 2d+2 , the central limit theorem of Proposition 4.1 can now be stated as

(4.7) X -nk σ n law ---→ n→∞ N (0, ∇ 2 Λ(u)).
The proof of the next proposition will use this fact.

Proposition 4.2 (Weak local limit estimate).

P n [X = nk] = e -o(n d/(d+1) )
Proof. We begin with the following simple remark: ,m). This explains the introduction of the following set,

if m ∈ Z d ∩ C and n ∈ Z d ∩ C satisfy n -m ∈ C, then p(C, n) ≥ p(C
A n = {m ∈ Z d ∩ C | nk -m ∈ C and u • (nk -m) ≤ σ n }.
As a consequence of (4.7), there exists c > 0 such that P n [X ∈ A n ] ≥ c for all n large enough. In addition, equation (4.3) and the remark above imply that

P n [X ∈ A n ] = m∈An p(C, m)e -βnu•m = m∈An p(C, m) p(C, nk) e βnu•(nk-m) P n [X = nk] ≤ |A n |e βnσn P n [X = nk].
Since σ n = n (d+2)/(2d+2) , the factor |A n | = O(σ d n ) grows only as a power of n while e βnσn = e O(n d/(2d+2) ) . Since P n [X ∈ A n ] ≥ c, the result is proven.

The number of generators of T ∈ T (C, nk). Zonotopes in T (C, nk)

have been identified with functions ω : P d ∩ C → Z + with finite support. The generators of T ∈ T (C, nk) are those x ∈ P d for which ω(x) > 0, we write G(T ) for the set of generators of T . Using the method of the previous subsection we determine the expected number µ n of the number of generators of T ∈ T (C, nk), under the distribution P n . By Proposition 4.2, it will also lead to an estimate under the uniform distribution Q n on T (C n k). Lemma 4.3.

µ n = E[|G(T )|] = 1 ζ(d)β d n Λ C (u)(1 + O(β n )).
Proof. Let β = β n in the proof. Formula (4.4) implies that

P n [ω(x) > 0] = e -βu•x . A consequence of Proposition A.1 is that x∈Z d ∩C e -βu•x - 1 ζ(d) C e -βu•x dx ≤ c d β d-1 ,
for some constant c d > 0 depending on the dimension. Here d+1) follows from the definition of β, the order of magnitude of µ is n d/(d+1) .

C e -βu•x dx = 1 β d C e u•x dx = 1 β d Λ C (u), and so µ = 1 ζ(d)β d Λ C (u)(1 + O(β)). As β -d n = Θ n d/(
We will need a similar result for the number of generators lying in a hyperplane H. Lemma 4.4. Assume H is a hyperplane in R d containing the origin. Then

µ H = E[|G(T ) ∩ H|] = O(n (d-1)/(d+1) ),
where the implicit constant in O(•) is independent of H.

Proof. Let β = β n in the proof. Following the above argument, we have

µ H = E[|G(T ) ∩ H|] = x∈Z d ∩C∩H e -βu•x .
The one-sided estimate of Lemma A.4 gives now that

x∈Z d ∩C∩H e -βu•x ≤ ζ(d) -1 C∩H e -βu•x dx + c d-1 β d-2 .
Here

C∩H e -βu•x dx = 1 β d-1 C∩H e u•x dx.
Thus indeed

µ H ≤ 1 + o(1) ζ(d)β d-1 C∩H e -u•x dx = O(n (d-1)/(d+1) ).
4.4. Limit shapes. In this section, we are going to show the existence of a limit shape for random zonotopes drawn under the uniform distribution on the set of all lattice zonotopes in the cone C with endpoint nk.

For every convex compact subset A of R d , the support function of A is the continuous function h A : R d → R defined by

h A (w) = sup{v • w, v ∈ A}, w ∈ R d .
Let T 0 = T 0 (C, k) be the zonoid defined in Section 3.

Theorem 4.5. For all v ∈ R d and for all ε > 0,

lim n→+∞ Q n |h 1 n T (v) -h T 0 (v)| > ε = 0. Proof.
Again, the proof of this theorem is based on the probabilistic model of Section 4. It depends on the fact that, conditional on the event X = nk, the distribution of a random zonotope under P n is uniformly distributed on the set T (C, nk). In particular,

Q n |h 1 n T (v) -h T 0 (v)| > ε = P n |h 1 n T (v) -h T 0 (v)| > ε | X = nk ≤ P n |h 1 n T (v) -h T 0 (v)| > ε P n [X = nk] .
Hence, we only need to prove that the right-hand side of this inequality goes to 0. It is natural to consider only v ∈ S d-1 , the unit sphere of R d . Using Proposition 4.2, we see that the theorem follows from the result of the next proposition.

Proposition 4.6. For all ε > 0 there exists c > 0 such that for all v ∈ S d-1 and for all n large enough,

P n |h 1 n T (v) -h T 0 (v)| > ε ≤ exp{-c n d/(d+1) }. Proof.
It is evident that for a zonotope T (ω) generated by the vectors x ∈ P d ∩ C with multiplicities ω(x) the support function is

h T (v) = x∈P d ∩C ω(x) x • v 1 {x•v≥0} = x∈P d ∩C(v≥0) ω(x) x • v.
In the proof of Proposition 4.1, we could replace C with the smaller cone C(v ≥ 0) and we would obtain

1 n E n [h T (v)] = Λ C(v≥0) (u) • v + O(n -1/(d+1) ) = h T 0 (v) + o(1).
Since 1 n h T = h 1 n T , this already shows that the limit shape appears in expectation.

We will now bound the probability for a large deviation from the mean using the so-called Chernoff method. Let us consider the exponential generating function of h T (v)

E[e θh T (v) ] = x∈P d ∩C(v≥0) 1 -e -βu•x 1 -e -(βu-θv)•x = Z C(v≥0) (βu -θv) Z C(v≥0) (βu)
which is well defined for all θ small enough with respect to β. Consider now the centered random variable

Y = h 1 n T (v) -E[h 1 n T (v)
]. The central limit part of Proposition 4.1 applies to the cone C(v ≥ 0) with k = -∇Λ C(v≥0) (u). Thus by second order Taylor approximation, we obtain for some constant c(u, v) > 0 involving the Hessian matrix of Λ C(v≥0) at u,

log E[e θY ] ∼ c(u, v) θ 2 2 n -d/(d+1)
as long as θ n goes to 0. But for all θ > 0, the Markov inequality yields

P n [Y > ε] = P n [θY ≥ θε] ≤ e -θε E[e θY ]
This bound is approximately optimized for θ = c(u, v) -1 ε n d/(d+1) and it leads to

P n [Y > ε] ≤ exp - 1 2 c(u, v) -1 ε 2 n d/(d+1
n d d+1 ---→ n→∞ c d q(C, k),
where

c d = d+1 ζ(d + 1) ζ(d) (d + 1)!
and where q(C, k) is defined in Section 3 as the minimal volume of a cap of the cone C having k as its center of gravity. The starting point of the proof is the equation (4.3) established earlier which links the number log p(C, k) with the probability distribution P n in the following way:

log p(C, k) = log Z n + nβ n u • k + log P n [X = nk].
The value of β n was given explicitly by (4.2) and it shows that nβ n is of order n d/(d+1) . On the other hand, we know by Proposition 4.2 that log P n [X = nk] is negligible compared to n d/(d+1) .

It only remains to estimate the term log Z n . This is done by using the series expansion (4.5) and applying Proposition A.1 from the appendix in the same lines as the first part of the proof of Proposition 4.1. This leads to

log Z n = ζ(d + 1) ζ(d) 1 β d n (Λ(u) + o(1)),
which is again of order n d/(d+1) again. A simple computation based on the definition of β n and of the constant c d yields therefore log p(C, nk)

n d d+1 ---→ n→∞ c d d+1 (d + 1)! (Λ(u) + u • k).
The final step is to express Λ(u) + u • k in terms of q(C, k). Recall that u is defined by (4.2) and that we have shown in Proposition 4.1 that

k = (d + 1)! C(u≤1) x dx = C xe -u•x dx.
The computation of Λ(u) + u • k is thus simply a matter of integral calculus. Using the fundamental theorem of calculus and the Fubini-Tonelli theorem, we obtain:

Λ(u) + u • k = C (1 + u • x)e -u•x dx = C ∞ u•x t e -t dt dx = ∞ 0 te -t Vol C(u ≤ t) dt = (d + 1)! Vol C(u ≤ 1)
Finally, the definition (4.2) of u and the considerations of volumes in Section 3 show that

Vol C(u ≤ 1) = q(C, 1 (d+1)! k) = 1 (d + 1)! d d+1 q(C, k).
The proof is complete.

Remark. Although we did not use it directly, it appears above that u is actually the unique minimizer of the quantity Λ(v) + v • k for v ∈ C • . This fact can be used to prove Gigena's Theorem 3.1.

Number of vertices of a random zonotope

In this section we determine the order of magnitude of the expected number of faces of each dimension and towers of a random zonotope. A tower (or flag) of a zonotope T is a chain

F 0 ⊂ F 1 ⊂ . . . ⊂ F d-1
, where F i is an i-dimensional face of T . We denote by f i (T ) the number of i-dimensional faces and by F (T ) the number of towers of T . Theorem 5.1. There are constants c 1 , c 2 not depending on n such that

lim n→+∞ Q n T ∈ T (C, nk) | c 1 < f i (T ) n d(d-1)/(d+1) < c 2 = 1, lim n→+∞ Q n T ∈ T (C, nk) | c 1 < F (T ) n d(d-1)/(d+1) < c 2 = 1.
As Q n is the uniform distribution on T (C, nk) the above theorem says that, as n → ∞, for the overwhelming majority of the zonotopes in T (C, nk) the face numbers f i (T ) and F (T ) are of order n d(d-1)/(d+1) . It is shown in [START_REF] Bárány | The convex hull of the integer points in a large ball[END_REF][START_REF] Konyagin | A bound, in terms of its volume, for the number of vertices of a convex polyhedron when the vertices have integer coordinates[END_REF] that for any integer lattice polytope P with vol P > 0,

F (P ) = O (vol P ) (d-1)/(d+1) .
The same upper bound for each f i (P ) follows immediately. Combined with the previous result on the volume of a random zonotope, this establishes the upper bounds of Theorem 5.1. It only remains to show the lower bounds.

Generators of a random zonotope. Let

T ε (C, nk) consist of all zonotopes T ∈ T (C, nk) such that |G(T )| ≥ (1 -ε)ζ(d) -1 Λ C (βu),
and the number of generators of T that are contained in any single hyperplane is at most εΛ C (βu). Our goal in this section is to prove all except for a minute fraction of the zonotopes in T (C, nk) belong to T ε (C, nk).

Proposition 5.2. lim n→+∞ Q n [T ε (C, nk)] = 1.
Proof. This is quite easy. We rely on the two lemmas in Subsection 4.3 and on the following rough form of the Chernoff bounds: d+1) , and so an application of the Chernoff bound gives, that for any 0 < ε < 1

Proposition 5.3 (Chernoff bounds). Let X 1 , . . . , X n be independent Bernoulli random variables, let X = X 1 + . . . + X n , and let µ = E[X]. For any δ > 0, P[X ≥ (1 + δ)µ] ≤ e -δ 2 µ/3 , 0 < δ < 1, P[X ≥ (1 + δ)µ] ≤ e -δµ/3 , 1 ≤ δ, P[X ≤ (1 -δ)µ] ≤ e -δ 2 µ/2 , 0 < δ < 1. By Lemma 4.3 µ = E[|G(T )|] is of order n d/(
P n [|G(T )| ≤ (1 -ε)µ] ≤ e -ε 2 µ/2 = exp{-cn d/(d+1) }
where c > 0 is a constant independent of n.

Next, we bound the probability that a hyperplane H with 0 ∈ H contains more than a constant fraction of the expected number of generators of a zonotope in T (C, nk). According to Lemma 4.4, µ H = E[|G(T ) ∩ H|] ≤ cn (d-1)/(d+1) with c not depending on H and n. Applying the Chernoff bound with γ > 1, we have

P n [|G(T ) ∩ H| ≥ (1 + γ)µ H ] ≤ e -γµ H /3 .
In particular,

P n [|G(T ) ∩ H| ≥ εΛ C (βu)] ≤ exp{-cn d/(d+1) }
for some c > 0 depending on ε.

Note finally that the total number of hyperplanes (containing the origin) spanned by the generators of a zonotope with O(n d/(d+1) ) generators is bounded by O(n d(d-1)/(d+1) ). Since this is polynomial in n, a union bound, together with Proposition 4.9, proves Proposition 5.2.

Faces of a random zonotope.

In this section, we show that the zonotopes in T ε (C, nk) have at least the number of i-dimensional faces prescribed by Theorem 5.1; the same lower bound for towers is an immediate corollary.

We rely on the following well-known duality between zonotopes and hyperplane arrangements. An arrangement of hyperplanes is a finite collection of affine hyperplanes, together with their subdivision of R d into relatively open cells. A central hyperplane arrangement is the decomposition of R d induced by linear hyperplanes. For a d-dimensional zonotope T with generators {v 1 , . . . , v m }, let A c (T ) be the central arrangement of hyperplanes with this set of normal vectors. The number of k-dimensional cells of A c (T ) is equal to the number of d -k faces of T (see, e.g., [6, Prop. 2.2.2]). Let A(T ) be the (d -1)-dimensional arrangement of affine hyperplanes obtained as the intersection of A c (T ) with a generic hyperplane. Clearly, the number of (k -1)-dimensional cells of A(T ) is a lower bound on the number of kdimensional cells of A(T ) for k ∈ [1, d]. We note that twice this lower bound is an upper bound on the number in question as the cells of A c (T ) come in pairs C, -C except C = {0}. Proposition 5.4. There is a constant c independent of n such that for all ε ∈ (0, c) and T ∈ T ε (C, nk), we have f i (T ) = Ω(n d(d-1)/(d+1) ) for each

0 ≤ i ≤ d -1.
Proof. If a single line {λv | λ ∈ R} is incident to k hyperplanes of A c (T ), then the vector v is orthogonal to k generators of T . Since at most (ε/(1ε))|G(T )| of the generators of T lie in the hyperplane v ⊥ (for any v), the same bound applies to the maximum number of hyperplanes of A(T ) that are incident to any single point. Hence, we can apply the following theorem of Beck [START_REF] Beck | On the lattice property of the plane and some problems of dirac, motzkin and erdos in combinatorial geometry[END_REF] to bound the number of vertices of A(T ).

Lemma 5.5.

There is a constant c depending on dimension such that the following holds. Let A be an arrangement of m hyperplanes in R d-1 . Then either

(1) a single vertex of A is incident to cm hyperplanes of A, or (2) the total number of vertices in A is Ω(m d-1 ).

Since A(T ) is an arrangement of n d/(d+1) hyperplanes, this implies that, for ε sufficiently small, the number of vertices of A(T ) is Ω(n d(d-1)/(d+1) ).

Designate a generic direction w in R d-1 to be "up". Clearly, each (d -1)dimensional region of A(T ) that is bounded from below has a bottom vertex. We claim that each vertex of A(T ) is the bottom vertex of at least one region. Let p be an arbitrary vertex of A(T ). At least d -1 hyperplanes having linearly independent normals are incident to p. Let S be an arbitrary set of d -1 such hyperplanes. At least one (d -1) dimensional region R of A(T ) is above each of these hyperplanes with respect to w, and p is the bottom vertex of each such region. In addition, p is the bottom vertex of each j-dimensional region that is contained in a hyperplane of S and that bounds R, for 1 ≤ j ≤ d -2. Hence, the number of j-dimensional regions that are bounded below in A(T ) is at least the number of vertices of A(T ), which is Ω(n d(d-1)/(d+1) ).

Since the j-dimensional regions of A(T ) are in bijection with the (d -j)dimensional faces of T , this completes the proof of the proposition.

A short digression.

This proof method has an interesting consequence about hyperplane (or rather subspace) arrangements that seems to be new. Recall first that if A is a central arrangement of m general position hyperplanes in R d , then f i (A) is known precisely for all i: Buck [7] and Zaslavski [START_REF] Zaslavsky | Facing up to arrangements: face-count formulas for partitions of space by hyperplanes[END_REF]. Moreover, this function is an upper bound on the number of i-dimensional cells of every central arrangement of m linear hyperplanes in R d . Define A c r as the collection of all d -1-dimensional subspaces of the form z ⊥ where z ∈ Z d has Euclidean length at most r. We could not find the following result anywhere in the literature or in folklore.

f i (A) = k k=d-i k d -i m k see for instance
Theorem 5.6. The number of i-dimensional cells for all i = 1, . . . , d and the number of towers of A r is Θ(r d(d-1) ).

Proof. It follows from the result of Beck [START_REF] Beck | On the lattice property of the plane and some problems of dirac, motzkin and erdos in combinatorial geometry[END_REF] cited above that the number of one-dimensional cells of A c r is Θ(r d (d-1) ). The same estimate is also implied by Theorem 3 of [START_REF] Bárány | Covering lattice points by subspaces[END_REF].

Let A r be the (d -1)-dimensional arrangement of affine hyperplanes obtained as the intersection of A c r with a general position hyperplane H the same way as above. The previous proof applies word by word.

Integral zonotopes in convex bodies

As it is mentioned in the introduction, for d = 2 Theorem 1.1 follows from the results of Bárány [START_REF] Bárány | Affine perimeter and limit shape[END_REF]. In fact, more is true. We set up the question more generally. Assume K ⊂ R d a convex body (i.e., convex compact set with non-empty interior) and write P(K, n) for the collection of all convex 1 n Z d -lattice polytopes contained in K, P(K, n) is a finite set. It is proved in [START_REF] Bárány | Affine perimeter and limit shape[END_REF] that, when d = 2, the polygons in P(K, n) have a limit shape as n → ∞: Theorem 6.1. For every convex body K in R 2 there is a convex body K 0 ⊂ K such that, as n → ∞, the overwhelming majority of the polygons in P(K, N ) are very close to K 0 . More precisely for every ε > 0

lim n→∞ card {P ∈ P(K, n) | dist(P, K 0 ) > ε} card P(K, n) = 0.
The distinguishing property of K 0 is that its affine perimeter (for the definition see [START_REF] Bárány | Affine perimeter and limit shape[END_REF]) is maximal among all convex subsets of K. It is also shown there that such a K 0 is unique. The analogous question in higher dimension is wide open. Even the case when K is the cube [-1, 1] 3 is not known. Question 1. Assume K ⊂ R d is a convex body and d > 2. Is there a limit shape to the convex polytopes in P(K, n)? Define, similarly, F(K, n) as the collection of all convex 1 n Z d -lattice zonotopes contained in K, F(K, n) is again a finite set. Question 2. Assume K ⊂ R d is a convex body and d ≥ 2. Is there a limit shape to the convex zonotopes in F(K, n)?

We can answer this question when d = 2. Informally stating, for every convex body K in the plane there is a zonoid K 0 contained in K such that, as n → ∞, the overwhelming majority of the polygons in F(K, n) are very close to K 0 . The proof follows the method of [START_REF] Bárány | Affine perimeter and limit shape[END_REF] so we only give a sketch. Note that in the plane a zonoid is always a centrally symmetric convex body and vice versa. The first thing to show is that K contains a centrally symmetric convex body that maximizes the affine perimeter among all centrally symmetric convex subsets of K. This follows from the fact that the affine perimeter is upper semi-continuous. The next step is to prove that the maximizer is unique. Here one uses the fact that, for all t ∈ [0, 1], if A, B ⊂ K are centrally symmetric convex bodies, then so is tA + (1 -t)B which is a subset of K, again. The affine perimeter is a concave functional, that is, writing AP (S) for the affine perimeter of the convex body S we have

AP (tA + (1 -t)B) ≥ tAP (A) + (1 -t)AP (B).
This shows that if A and B are both maximizers, then so is tA + (1 -t)B for all t ∈ [0, 1]. Next one proves that A ⊂ B cannot happen. Then A and B are both maximizers not only for K but also for the convex body Conv(A ∪ B). Then A = B follows more or less the same way as in [START_REF] Bárány | Affine perimeter and limit shape[END_REF]. The proof that the unique maximizer is the limit shape in P(K, n) goes analogously to [START_REF] Bárány | Affine perimeter and limit shape[END_REF].

We can also answer Question 2 when the convex body K is the cube

Q = Q d = [-1, 1] d for all d ≥ 2.
To describe the limiting zonoid in this case, set t = d+1 (d + 1)!2 1-d and define the octahedron

O d = t Conv{±e 1 , . . . , ±e d }.
With this notation the support function of the limiting zonoid Q 0 is given by

h Q 0 (v) = 1 2 O d |x • v|dx.
Theorem 6.2. With the above notation for every d ≥ 2 and for every η > 0 lim Extend the cone with sign pattern ε to another cone, to be denoted by C ε so that these 2 d cones form a partition of R d . This is clearly possible.

n→∞ card P ∈ F(Q d , n) | dist(P, Q 0 ) > η card F(Q d , n) = 0.
As we have seen in Section 2 for every zonotope T ∈ P(Q, n) there is a unique multiset W ⊂ 1 n P d such that T = T (W ). Define W ε = W ∩ C ε for every ε and set k ε = w∈W ε w. Then T ε = T (W ε ) is a zonotope in the cone C ε with endpoint k ε and containing the origin. Moreover, T = ε T ε . As T ⊂ Q, the conditions (6.1)

ε:ε i =1 k ε i ≤ 1 and ε:ε i =-1 k ε i ≥ -1 are satisfied for every i = 1, . . . , d. A fixed k ε ∈ C ε ∩ 1 n Z defines a family F(k ε
) consisting of all zonotopes in C ε that contain the origin and whose endpoint is k ε . Given k ε ∈ C ε for all ε we write k = (k ε : all ε) which is in fact a d × 2 d matrix. Let F(k) be the collection of all zonotopes T ∈ F(Q, n) for which the endpoint of T ε is k ε for all ε.

After these preparation here is the plan of the proof. We show that the contribution to

|F(Q, n)| of F(k) is minute unless |k ε i -ε i 2 1-d
| is very small for all ε and all i. This will prove that for most zonotopes T ∈ F(Q, n) and for all ε the endpoint of T ε is very close to the vector (ε 1 2 1-d , . . . , ε d 2 1-d ). The limit shape of these zonotopes is very close to the limiting zonoid of the zonotopes in C ε with endpoint (ε 1 2 1-d , . . . , ε d 2 1-d ) which is known from Example 1. Finally, the sum of the limiting zonoids is Q 0 as one can check easily.

It follows from Theorem 1.2 and Example 1 that

log |F(k ε )| = c d (d + 1) d d! d 1 |k ε i | 1/(d+1) n d/(d+1) (1 + o(1)).
Since log |F(k)| = ε log |F(k ε )| we will have to deal with the expression

S(k) := ε d 1 |k ε i | 1/(d+1)
.

Writing c d = c d d+1 (d+1) d d! , we have log F(k) = c d S(k)n d/(d+1) (1 + o(1)).
Our first task is to show that (6.2)

S(k) ≤ d d + 1 2 2d/(d+1) (1 + 2 1-d ) =: D d .
To see this it is convenient to define k ε 0 = 2 1-d . Then, using the inequality between the arithmetic and geometric means 2 d times and the conditions (6.1)

S(k) = 2 (d-1)/(d+1) ε d 0 |k ε i | 1/(d+1) ≤ 2 (d-1)/(d+1) 1 d + 1 ε d 0 |k ε i | = 2 (d-1)/(d+1) 1 d + 1 d 0 ( ε |k ε i |) ≤ 2 (d-1)/(d+1) 2 d + 1 d 0 (1 + 2 1-d ) = d d + 1 2 2d/(d+1) (1 + 2 1-d ) = D d .
Note that equality holds throughout if all k ε i = ±2 1-d , so max S(k) = D d . Here k ε can be chosen in at most (n + 1) d ways, so k can take at most (n + 1) d2 d values. Then

|F(Q, n)| = k exp{c d S(k)n d/(d+1) (1+o(1))} ≤ exp{c d D d n d/(d+1) (1+o(1))}.
Considering the maximal term in the above sum it follows that

|F(Q, n)| = exp{c d D d n d/(d+1) (1 + o(1))}.
We are to use a strengthening of the inequality between the arithmetic and geometric means. Let t 1 , . . . , t m be non-negative reals (not all zero) and let A m resp. G m denote their arithmetic and geometric mean. Then, for any i, j in {1, . . . , m}

(6.3) (t i -t j ) 2 2m 2 A m ≤ A m -G m .
For the reader's convenience we prove this inequality at the end of this section. Assume next that |k ε j -ε j 2 1-d | ≥ δ for some ε and for some j = 1, . . . , d where δ > 0. Then using (6.3) in the above application of the inequality between the arithmetic and geometric means in the term corresponding to this ε we have can add an extra term, namely

( d 0 |k ε i |) 1/(d+1) ≤ 1 d + 1 ε ( d 0 |k ε i |) - (k ε j -ε j 2 1-d ) 2 2(d + 1) 2 (1 + 2 1-d ) ≤ 1 d + 1 ε ( d 0 |k ε i |) - δ 2 3(d + 1) 2 .
Here we used the fact that d j=0 k ε j ≤ 1 + 2 1-d . This implies that when

|k ε i -ε i 2 1-d | ≥ δ for some positive δ, then S(k) ≤ D d - δ 2 3(d + 1) 2 .
This shows in turn that the contribution of the F(k) with all such k to d+1) is larger than any fixed positive constant, say η 2 . Thus, for all δ > 0, as n → ∞, all but a small fraction of the zonotopes in F(Q, n) satisfy |k ε i -ε i 2 1-d | ≤ δ for all ε and all i. Theorem 1.1 and Example 1 show that, again, all but a small fraction of the zonotopes in F(k ε ) are very close to the limiting zonoid in

|F(Q, n)| is only exp{c d (D d -δ 2 /3(d + 1) 2 )n d/(d+1) (1 + o(1))}, much smaller than |F(Q, n)| as long as δ 2 n d/(
C ε corresponding to k ε = (ε 1 2 1-d , . . . , ε d 2 1-d ).
Proof of (6.3). We may assume that i = 1 and j = 2. Suppose m ≥ 3 and let A m-1 resp. G m-1 denote the arithmetic and geometric mean of t 1 , . . . , t m-1 . Richard Rado [10, Theorem 60] proved the inequality

(m -1)(A m-1 -G m-1 ) ≤ m(A m -G m ),
which is, of course, a strengthening of the inequality between the arithmetic and geometric means. Then for m ≥ 2

m(A m -G m ) ≥ 2(A 2 -G 2 ) = ( √ t 1 - √ t 2 ) 2 = (t 1 -t 2 ) 2 ( √ t 1 + √ t 2 ) 2 ≥ (t 1 -t 2 ) 2 2(t 1 + t 2 ) ≥ (t 1 -t 2 ) 2 2mA m .
We start the proof by a standard lemma dealing with the approximation of integrals on a convex body by Riemann sums. Proof. For all x in Z d , let us consider the (hyper)cube Q(x) = x + [-1 2 , 1 2 ] d of unit volume. These cubes are of three types: those that are contained in K, those that cross the boundary of K, and those that have no point in common with K. The idea of the proof is to approximate the integral by considering in first approximation only the reunion of the cubes of the first kind. For each cube Q(x) contained in K, f (x) -

Q(x) f (y)dy ≤ Q(x) |f (x) -f (y)| dy ≤ M √ d 2 .
Moreover, the number of cubes contained in K is at most L d . This already yields the first term in the right-hand side of (A.2).

We also have to deal with the cubes that cross the boundary of K. Since K is convex, it is easily seen by induction on d that the number of such cubes is at most 2d!(L + 1) d-1 . Bounding f by sup K |f | on Q(x) ∩ K, we obtain (A.2). Proof. For x ∈ Z d ∩ H, let Q(x) = x + [-1/2, 1/2] d be the hypercube of unit volume centered at x. A theorem of Vaaler [START_REF] Vaaler | A geometric inequality with applications to linear forms[END_REF] establishes that the (d -1)-dimensional volume of H ∩ Q(x) is at least 1; let Q (x) be a subset of Q(x) of volume 1.

For Q (x) contained in K, f (x) -

Q (x) f (y)dy ≤ Q (x) |f (x) -f (y)| ≤ M √ d/2.
The other error term comes from a bound on the number of cubes that intersect the boundary of K, and the proof is exactly as in the proof of Lemma A.2.

We do not have a two-sided inequality as in Lemma A.2, since the union of Q does not contain the intersection K ∩ H, and we have also lost some of the mass by passing from Q to Q .

Theorem 1 . 1 .

 11 Given C ∈ C d (d ≥ 2) and k ∈ Int C there is a convex set T 0 = T 0 (C, k) such that for every ε > 0 lim n→∞ card T ∈ T (C, nk) | dist( 1 n T, T 0 ) > ε p(C, nk) = 0.

Example 1 ..Example 2 .3 π 3 4 1/ 4 = 1 .

 12441 Let C = R d + = Pos{e 1 , . . . , e d } be the positive orthant of R d where the e i form the standard basis of R d , and a = (a 1 , . . . , a d ) ∈ Int C. Then the section C(u = 1) is the intersection of C with the hyperplane passing through the points da i e i , (i = 1, . . . , d), and Vol C(u ≤ 1) = 1 d! d d a 1 . . . a d . Consequently q(C, a) = (d + 1) d d! a 1 . . . a d 1/(d+1) Let C be the circular cone in R 3 of equation x 2 + y 2 ≤ z 2 with z ≥ 0 and consider k = (0, 0, 1). The minimal cap of C containing k is the one cut off by the plane z = 1. Its volume is π/3, so in this case q(C, k) = 4 255294... Next we explain what the limiting zonoid T 0 = T 0 (C, k) from Theorem 1.1 is. Its support function is given by

Figure 1 .

 1 Figure 1. The limiting zonoid in dimension 2.

Figure 2 .

 2 Figure 2. The limiting zonoid in dimension 3.

x∈Vnx

  = nk(1 + o(1)). Suppose now that x∈S x = nk for some S ⊂ Z d ∩C. Under these conditions Proposition 3.2 implies that |S| ≤ |V n |(1 + o(1)).

Proof.

  The coordinate hyperplanes x i = 0 for i = 1, . . . , d split R d into 2 d open cones; each corresponds to a sign pattern ε = (ε 1 , . . . , ε d ) ∈ {-1, 1} d .

Lemma A. 2 .

 2 Let L > 0, and let K be a compact convex subset of the hypercube [-L 2 , L 2 ] d . For every Lipschitz continuous function f : K → R with Lipschitz constant M > 0,

Corollary A. 3 . 1 .Lemma A. 4 .

 314 Let f : C → R be continuously differentiable and positively homogeneous of degree h, that is to say f (λx) = λ h f (x) for all x ∈ C and λ ≥ 0. Let A be a compact subset of C • . There exists c A,f,d > 0 such that for all β ∈ (0, 1],(A.3) sup u∈A x∈C∩Z d f (x)e -βu•x -C f (x)e -βu•x dx ≤ c A,f,d β d+h-1 .Proof. Since A is compact, we can find L > 0 such that for all u ∈ A, the truncated coneC u = {x ∈ C | u • x ≤ 1} is contained in [-L 2 , L 2 ] d . For all t > 0, an application of Lemma A.2 to the compact convex subset tC u of [-tL 2 , tL 2 ] d implies sup u∈A x∈tCu∩Z d f (x) -tCu f (x)dx f,A,d (1 + t) d+h-By integration over [0, +∞), we obtain therefore for all β > 0dx βe -βt dt f,A,d1 β d+h-1 .This concludes the proof since the Fubini theorem yields∞ 0   x∈C∩Z d f (x)1 {u•x≤t}   βe -βt dt = x∈C∩Z d f (x)e -βu•x and similarly ∞ 0 tCu f (x)dx βe -βt dt = x∈C f (x)e -βu•x .Proof of Proposition A.1. This follows from Corollary A.3 and the fact that the subset P d of Z d has asymptotic density 1/ζ(d), which is a well known consequence of the Möbius inversion formula[START_REF] Hardy | An introduction to the theory of numbers[END_REF].Here comes the one-sided inequality needed in the proof of Theorem 5.1. Let L > 0, let H be a hyperplane containing the origin. Let K be a compact convex subset of the hypercube [-L/2, L/2] d ∩ H. For every Lipschitz continuous function f : K → R with Lipschitz constant M > 0,x∈K∩Z d f (x) -K f (x)dx ≤ M √ d 2 L d-1 + 4(d -1)!(L + 1) d-2 sup K |f |.

Acknowledgments

The authors are pleased to thank Nathanaël Enriquez for his insightful suggestions about this work. This research was supported by ERC Advanced Research Grant no 267165 (DISCONV). Imre Bárány is partially supported by Hungarian National Research Grant K 111827. Ben Lund is partially supported by NSF grant CCF-1350572.

Appendix A. Discrete sums and integrals on cones

The aim of this section is to establish the following proposition.