Saliou Cheikh

Ndiaye

A NOTE ON BÉZOUT TYPE INEQUALITIES FOR MIXED VOLUMES AND MINKOWSKI SUMS

Keywords: Mathematics Subject Classification: 52A39, 52A40 Mixed volume, Minkowski sum, Convex body, optimal transport, Alexandrov-Fenchel, Bézout inequality

In this note, we study Bézout type inequalities for mixed volume and Minkowski sum of convex bodies in R n . We first give a new proof and we extend inequalities of Jian Xiao on mixed discriminants. Then, we use mass transport method to deduce some Bézout type inequalities for mixed volumes. Finally, we apply these inequalities to obtain Bézout type inequalities for Minkowski sums.

Introduction

In 1779, Bézout gave the first version of his theorem about the number of intersection points of algebraic hypersurfaces having no component in common: if H 1 , . . . , H n are algebraic hypersurfaces in C n , then

#(H 1 , . . . , H n ) ≤ deg(H 1 ) × • • • × deg(H n ). (1)
From the Bernstein-Kushnirenko-Khovanskii theorem (see [START_REF] Bernstein | The number of roots of a system of equations[END_REF][START_REF] Khovanskii | Newton polyhedra and the genus of complete intersections[END_REF][START_REF] Kushnirenko | Newton polyhedra and Bézout's theorem (Russian) Funkcional[END_REF]), the quantities appearing in this inequality may be written in terms of mixed volumes of convex bodies. For any K 1 , . . . , K m convex bodies in R n , their mixed volume is defined as

V (K 1 , . . . , K n ) = 1 n! n k=1 (-1) n+k i 1 <•••<i k |K i 1 + • • • + K i k |,
where

K i 1 + • • • + K i k := {x i 1 + • • • + x i k |x i j ∈ K i j }
is the Minkowski sum and | • | is the Lebesgue measure. Minkowski showed that for t 1 , . . . , t m ∈ R + , |t 1 K 1 + • • • + t m K m | can be extended as a polynomial function (see section 5.1 of [START_REF] Schneider | Convex bodies: the Brunn-Minkowski theory[END_REF]):

|t 1 K 1 + • • • + t m K m | = |i|=n n! i 1 !i 2 ! • • • i m ! V (K 1 [i 1 ], . . . , K m [i m ])t i 1 1 • • • t im m ,
where K[i j ] is K taken i j times. Soprunov and Zvavitch showed in [START_REF] Soprunov | Bézout inequality for mixed volumes[END_REF] that the inequality (1) can be rewritten in the following way: for 2 ≤ r ≤ n, and for all P 1 , . . . , P r convex bodies,

|∆| r-1 V (P 1 , . . . , P r , ∆[n -r]) ≤ r i=1 V (P i , ∆[n -1]), (2)
where ∆ is a n-dimensional simplex. If H 1 , . . . , H r are generic hypersurfaces in (C * ) n , P 1 , . . . , P r their Newton polytope and H r+1 , . . . , H n are generic linear forms, then the standard simplex conv{0, e 1 , . . . , e n } is the Newton polytope of H r+1 , . . . , H n and by the Bernstein-Kushnirenko-Khovanskii theorem, #(H 1 ∩ • • • ∩ H n ) = n!V (P 1 , . . . , P r , ∆[n -r]), for all i ∈ {1, . . . , r}, deg(H i ) = n!V (P i , ∆[n -1]) and for all i ∈ {r + 1, . . . , n}, deg(H i ) = 1. Therefore, in this case, the inequality (1) becomes (2) by replacing n! by 1/|∆|. Soprunov and Zvavitch [START_REF] Soprunov | Bézout inequality for mixed volumes[END_REF] conjectured that if A satisfies

|A| r-1 V (B 1 , . . . , B r , A[n -r]) ≤ r i=1 V (B i , A[n -1]),
This research is partly funded by the Bézout Labex, funded by ANR, reference ANR-10-LABX-58.

for all convex bodies B 1 , . . . , B r , then A is an n-dimensional simplex and some positive partial answers are given in [START_REF] Saroglou | Characterization of simplices via the Bézout inequality for mixed volumes[END_REF][START_REF] Soprunov | Bézout inequality for mixed volumes[END_REF][START_REF] Szusterman | A new condition towards the Soprunov-Zvavitch on Bézout-type inequalities[END_REF]. In [START_REF] Soprunov | Bézout inequality for mixed volumes[END_REF], an inequality in the same flavour where A can be an arbitrary convex body was also studied and the question of the best constant b n,r such that for all convex bodies A, B 1 , . . . , B r in R n ,

|A| r-1 V (B 1 , . . . , B r , A[n -r]) ≤ b n,r r i=1 V (B i , A[n -1]) (3)
was considered. Inspired by the works of Fradelizi, Giannopoulos, Hartzoulaki, Meyer, and Paouris in [START_REF] Fradelizi | Some inequalities about mixed volumes[END_REF][START_REF] Giannopoulos | On a local version of the Aleksandrov-Fenchel inequality about the quermasintegrals of a convex body[END_REF], it was proved in [START_REF] Soprunov | Bézout inequality for mixed volumes[END_REF] that b n,r ≤ (nr) r r! . Xiao used the inequality (7) below to show that b n,r ≤ n r-1 in [START_REF] Xiao | Bézout type inequality in convex geometry[END_REF]. Brazitikos, Giannopoulos and Liakopoulos [START_REF] Brazitikos | Uniform cover inequalities for the volume of coordinate sections and projections of convex bodies[END_REF] showed that b n,r ≤ 2 2 r-1 -1 . Our first main result is the following improvement of the preceding bounds. We show that b n,r ≤ min k∈{1,...,r}

2 k(k-1) 2 n r-k (r -k)! . (4)
In particular, for k = 1, this gives b n,r ≤ n r-1 (r-1)! and for k = r, we get b n,r ≤ 2 r(r-1) 2

. Our main tools are improved Xiao's argument and the following Fenchel's inequality ( [START_REF] Fenchel | Généralisation du théorème de Brunn et Minkowski concernant les corps convexes[END_REF], see also inequality (7.76) in [START_REF] Schneider | Convex bodies: the Brunn-Minkowski theory[END_REF]): [START_REF] Bernstein | The number of roots of a system of equations[END_REF] for any A, B, C, K 1 , . . . , K n-2 convex bodies in R n . We establish (4) by using mixed discriminants which are defined in the following way: for M 1 , . . . , M m be semi-definite positive symmetric matrices in R n and t 1 , . . . , t m ∈ R + .

V (A[2], K 1 , . . . , K n-2 )V (B, C, K 1 , . . . , K n-2 ) ≤ 2V (A, B, K 1 , . . . , K n-2 )V (A, C, K 1 , . . . , K n-2 ),
det (t 1 M 1 + • • • + t m M m ) = |i|=n n! i 1 !i 2 ! • • • i m ! D(M 1 [i 1 ], . . . , M m [i m ])t i 1 1 • • • t im m , the coefficient D(M 1 [i 1 ], . . . , M m [i m ]
) is the mixed discriminant. The mixed discriminants are non negative and for any linear invertible map T ,

D(T M 1 , . . . , T M n ) = det(T )D(M 1 , . . . , M n ), (6)
(see section 5.5 of [START_REF] Schneider | Convex bodies: the Brunn-Minkowski theory[END_REF]). In [START_REF] Xiao | Bézout type inequality in convex geometry[END_REF], Xiao proved the following inequality which we extend [START_REF] Brenier | Polar factorization and monotone rearrangement of vector-valued functions[END_REF] in theorem 1 below: for any integers k, n such that 1 ≤ k ≤ n and any positive definite symmetric matrices A, B, M 1 , . . . , M n-k , we have

det(A)D(B[k], M 1 , • • • , M n-k ) ≤ n k D(A[n -k], B[k])D(A[k], M 1 , • • • , K n-k ). (7)
Our second main theorem is the application of these new Bézout type inequality for mixed volumes to establish a new Bézout type inequality for Minkowski sums. Let c n,m be the smallest constant such that for any convex bodies

A, B 1 , . . . , B m ⊂ R n , |A| m-1 |A + B 1 + • • • + B m | ≤ c n,m m k=1 |A + B k |.
Bobkov and Madiman established in [START_REF] Bobkov | Reverse Brunn-Minkowski and reverse entropy power inequalities for convex measures[END_REF] that c n,m ≤ (m + 1) n . Fradelizi, Madiman and Zvavitch used a consequence of [START_REF] Brenier | Polar factorization and monotone rearrangement of vector-valued functions[END_REF] in [START_REF] Fradelizi | Sumsets estimates in convex geometry[END_REF] to show that 4 3 + o(1) n ≤ c n,2 ≤ ϕ n where ϕ is the golden ratio. They also gave more precise bounds for small dimensions: c 2,2 = 1 c 3,2 = 4/3 and they conjectured that c 4,2 = 3/2. There are several similar results for Minkowski sums of zonoids in [START_REF] Fradelizi | On the volume of the Minkowski sum of zonoids[END_REF].

In Section 2, we give an extension of inequality [START_REF] Brenier | Polar factorization and monotone rearrangement of vector-valued functions[END_REF] and then, we use optimal transport as introduced in [START_REF] Alesker | A remarkable measure preserving diffeomorphism between two convex bodies in R n[END_REF] and developed in [START_REF] Lehmann | Correspondences between convex geometry and complex geometry[END_REF][START_REF] Xiao | Bézout type inequality in convex geometry[END_REF] to deduce the same type of inequality for mixed volumes. In Section 3, we establish our Bézout inequality for mixed volumes (4) and we conclude with some Bézout type inequalities for Minkowski sums in Section 4.

Xiao type inequalities

Our first theorem is the following extension of inequality [START_REF] Brenier | Polar factorization and monotone rearrangement of vector-valued functions[END_REF] due to Xiao [START_REF] Xiao | Bézout type inequality in convex geometry[END_REF].

Theorem 1. Let m ≥ 1, n ≥ 2 and |i| := i 1 +• • •+i m ≤ n be integers. Then, for any semi-definite positive symmetric matrices A, B 1 , . . . , B m , M 1 , . . . , M n-|i| , |i|! i 1 ! . . . i m ! det(A) m D(B 1 [i 1 ], . . . , B m [i m ], M 1 , . . . , M n-|i| ) ≤ D(M 1 , . . . , M n-|i| , A[|i|]) m k=1 n i k D(B k [i k ], A[n -i k ]). ( 8
)
Proof: We do this proof by induction on m. The first step is a consequence of ( 7) of which we give a new proof inspired in ideas from [START_REF] Artstein-Avidan | Remarks about mixed discriminants and volumes[END_REF]. Case m = 1: (Xiao [START_REF] Xiao | Bézout type inequality in convex geometry[END_REF]) If m = 1, B 1 = B and i 1 = k ≤ n, (8) becomes [START_REF] Brenier | Polar factorization and monotone rearrangement of vector-valued functions[END_REF]. Since A and B 1 are semi-definite positive symmetric matrices, by simultaneous orthogonalization, there exists an invertible matrix P and a diagonal matrix Λ such that A = P P t and B 1 = P ΛP t . Using [START_REF] Bobkov | Reverse Brunn-Minkowski and reverse entropy power inequalities for convex measures[END_REF] it is enough to assume in (7) that A = I, B = diag(λ 1 , . . . , λ n ) and to prove that

D (B[k], M 1 , . . . , M n-k ) ≤ n k D (B[k], I[n -k]) D (I[k], M 1 , . . . , M n-k ) .
For any semi-definite positive symmetric matrices C 1 , . . . , C n , the polarization formula of the mixed discriminant is

D(C 1 , . . . , C n ) = 1 n! σ∈Sn det C 1 σ(1) , . . . , C n σ(n) ,
where C j i denotes the j -th column of C i (see section 5.5 of [START_REF] Schneider | Convex bodies: the Brunn-Minkowski theory[END_REF]). Taking

C 1 = • • • = C k = B and for i ∈ {k + 1, . . . , n}, C i = M i-k and expanding the discriminants, one has D(B[k], M 1 , . . . , M n-k ) = 1 n! |K|=k k! j∈K λ j σ∈Sn det {σ(1),...,σ(k)}=K M K c σ(k+1) 1 , . . . , M K c σ(n) n-k = 1 n k |K|=k j∈K λ j D(M K c 1 , . . . , M K c n-k ),
where for any K subset of [n] := {1, . . . , n} of cardinality k and any n-dimensional matrix M ,

M K c is the (n-k) dimensional matrix defined by M K c = (M i,j ) i,j∈[n] K and D(M K c 1 , . . . , M K c n-k ) their mixed discriminant in dimension n -k.
In the same way, we have,

D(I[k], M 1 , . . . , M n-k ) = 1 n k |K|=k D(M K c 1 , . . . , M K c n-k ) and D(I[k], B[n -k]) = 1 n k |J|=k j∈J λ j . It follows that n k D(I[n -k], B[k])D(I[k], M 1 , . . . , M n-k ) = 1 n k |J|=k j∈J λ j |K|=k D(M K c 1 , . . . , M K c n-k ) ≥ 1 n k |K|=k j∈K λ j (M K c 1 , . . . , M K c n-k ) = D(B[k], M 1 , . . . , M n-k ).
Hence, the case m = 1 is proved. Induction step: Let us assume that inequality ( 8) is true for m -1. We can assume that A = I, B m = diag(δ 1 , . . . , δ n ) and we need to prove that

(n -i m )! i 1 ! . . . i m-1 !(n -|i|)! |Jm|=im j∈Jm δ j D B J c m 1 [i 1 ], . . . , B J c m m-1 [i m-1 ], M J c m 1 , . . . , M J c m n-|i| ≤ |J|=|i| D(M J c 1 , . . . , M J c n-|i| ) m k=1 |J k |=i k det(B j k k ) = |Jm|=im j∈Jm δ j |J|=|i| D(M J c 1 , . . . , M J c n-|i| ) m-1 k=1 |J k |=i k det(B j k k ).
Thus, by comparing term by term, it is enough to prove that for each J m ,

(n -i m )! i 1 ! . . . i m-1 !(n -|i|)! D(B J c m 1 [i 1 ], . . . , B J c m m-1 [i m-1 ], M J c m 1 , . . . , M J c m n-|i| ) ≤ |J|=|i| D(M J c 1 , . . . , M J c n-|i| ) m-1 k=1 |J k |=i k det(B j k k ). ( 9
)
But the induction allows to say that, for such a fixed J m ,

(n -i m )! i 1 ! . . . i m-1 !(n -|i|)! D(B J c m 1 [i 1 ], . . . , B J c m m-1 [i m-1 ], M J c m 1 , . . . , M J c m n-|i| ) ≤ |J|=|i| J c ⊂J c m D(M J c 1 , . . . , M J c n-|i| ) m-1 k=1 |J k |=i k J k ⊂J c m det(B j k k ),
which implies [START_REF] Fradelizi | Some inequalities about mixed volumes[END_REF]. ✷ Remark 2. The method that we have used in the previous proof allows to answer positively to the Remark 3.6 in [START_REF] Lehmann | Correspondences between convex geometry and complex geometry[END_REF] where the author conjectured that for |i| ≤ n and for any semi-definite positive symmetric matrices A, B 1 , . . . , B m ,

det(A) m-1 D (B 1 [i 1 ], . . . , B m [i m ], A[n -|i|]) ≤ (n!) m-1 (n -|i|)! m j=1 (n -i j )! m k=1 D (B k [i k ], A[n -i k ]) . Theorem 3. Let m ≥ 1, n ≥ 2 and |i| := i 1 + • • • + i m ≤ n be integers. Then, for any convex bodies A, B 1 , . . . , B m , K 1 , . . . , K n-|i| in R n , |i|! i 1 ! . . . i m ! |A| m V (B 1 [i 1 ], . . . , B m [i m ], K 1 , . . . , K n-|i| ) ≤ V (K 1 , . . . , K n-|i| , A[|i|]) m k=1 n i k V (B k [i k ], A[n -i k ]). ( 10
)
Proof: We follow the method of Alesker, Dar and Milman [START_REF] Alesker | A remarkable measure preserving diffeomorphism between two convex bodies in R n[END_REF], introduced in this context by Lehmann and Xiao [START_REF] Lehmann | Correspondences between convex geometry and complex geometry[END_REF]. Let γ n be the gaussian measure on R n and for any convex body K, U (K) be the uniform distribution on K. By Brenier's theorem [START_REF] Brenier | Polar factorization and monotone rearrangement of vector-valued functions[END_REF][START_REF] Villani | Topics in Optimal Transportation[END_REF], for k = 1, . . . , m, there exists a convex function

f k such that ∇f k pushes forward γ n onto U (B k ). Then ∇f k (R n ) = B, |B k | = R n det(∇ 2 f k ), (11)
and for j = 1, . . . , n -|i|, there exists h j such that ∇h j pushes forward γ n onto U (K j ). It was proved in [START_REF] Alesker | A remarkable measure preserving diffeomorphism between two convex bodies in R n[END_REF] that if ∇g 1 and ∇g 2 are two Brenier's maps, then

∇g 1 (R n ) + ∇g 2 (R n ) = (∇g 1 + ∇g 2 )(R n ).
So, equality (11) also holds for Minkowski sums and therefore for mixed volume. Hence

V (B 1 [i 1 ], . . . , B m [i m ], K 1 , . . . , K n-|i| ) = R n ϕ(x)dx, where ϕ = D(∇ 2 f 1 [i 1 ], . . . , ∇ 2 f m [i m ], ∇ 2 h 1 , . . . , ∇ 2 h n-|i| ).
If ϕ = 0 the theorem is proved, so, we assume that ϕ = 0. Let µ be the probability measure having density dµ dx = 1 ϕ ϕ(x). There exists a convex function f A such that ∇f A pushes forward µ onto U (A). Since Brenier's map satisfies the Monge-Ampère equation (see Subsection 4.1.1 of [START_REF] Villani | Topics in Optimal Transportation[END_REF]), one has

|A| ϕ(x) ϕ = det(∇ 2 f A (x)) µ -almost everywhere in R n . ( 12
) Let ψ = D(∇ 2 f A [n -|i|]∇ 2 h 1 , . . . , ∇ 2 h n-|i|
), using Hölder inequality, we get that

n |i| V (K 1 , . . . , K n-|i| , A[|i|]) m k=1 n i k V (B k [i k ], A[n -i k ]) = n |i| R n ψ(x)dx m k=1 R n n i k D(∇ 2 f A (x)[n -i m ]∇ 2 f k (x)[i m ])dx ≥   R n n |i| ψ(x) m k=1 n i k D(∇ 2 f A (x)[n -i m ]∇ 2 f k (x)[i m ]) 1 m+1 dx   m+1 .
The inequality [START_REF] Fenchel | Généralisation du théorème de Brunn et Minkowski concernant les corps convexes[END_REF] in Theorem 1 allows to get

  R n n |i| ψ(x) m k=1 n i k D(∇ 2 f A (x)[n -i m ]∇ 2 f k (x)[i m ]) 1 m+1 dx   m+1 ≥ R n n! i 1 ! . . . i m !(n -|i|)! det ∇ 2 f A (x) m × ϕ(x) 1 m+1 dx m+1 = n! i 1 ! . . . i m !(n -|i|)! V (B 1 [i 1 ], . . . , B m [i m ], K 1 , . . . , K n-|i| ))|A| m ,
by taking into account the equality [START_REF] Giannopoulos | On a local version of the Aleksandrov-Fenchel inequality about the quermasintegrals of a convex body[END_REF].

✷ Corollary 4. Let 1 ≤ m ≤ n , |i| := i 1 + • • • + i m ≤ n be integers and A, B 1 , . . . , B m be convex bodies in R n .
Then, for any j ∈ {1, . . . , m},

(|i| -i j )!n! i 1 ! . . . i m !(n -i j )! V (B 1 [i 1 ], . . . , B m [i m ], A[n -|i|])|A| m-1 ≤ m k=1 n i k V (B k [i k ], A[n -i k ]). ( 13
)
Proof: Without loss of generality we can assume that j = m. Replacing m by m -1 in [START_REF] Fradelizi | On the volume of the Minkowski sum of zonoids[END_REF] gives

(|i| -i m )! i 1 ! . . . i m-1 ! |A| m-1 V (B 1 [i 1 ], . . . , B m-1 [i m-1 ], K 1 , . . . , K n-|i|+im ) ≤ V (K 1 , . . . , K n-|i|+im , A[|i|]) m-1 k=1 n i k V (B k [i k ], A[n -i k ]). ( 14
)
We obtain the desired result by taking [START_REF] Khovanskii | Newton polyhedra and the genus of complete intersections[END_REF] and by multiplying both sides by n im . ✷ Remark 5. Notice that Corollary 4 improves the following Xiao's inequality [START_REF] Xiao | Bézout type inequality in convex geometry[END_REF]: for any integers

K 1 = • • • = K im = B m in and K im+1 = • • • = K n-|i|+im = A in
2 ≤ m ≤ n, |i| := i 1 +• • •+i m ≤ n and convex bodies A, B 1 , . . . , B m in R n , for any j ∈ {1, . . . , m}, n i j V (B 1 [i 1 ], . . . , B m [i m ], A[n -|i|])|A| m-1 ≤ m k=1 n i k V (B k [i k ], A[n -i k ]).
(Similars results are given in [START_REF] Károly | Reverse Alexandrov-Fenchel inequalities for zonoids[END_REF] with A being the Euclidean ball.)

3. Bézout inequality for mixed volumes.

In this section, we present upper bounds on the constant b n,r defined in [START_REF] Artstein-Avidan | Remarks about mixed discriminants and volumes[END_REF]. Here, our first argument is Lemma 7 below which is a generalization of Fenchel's inequality [START_REF] Bernstein | The number of roots of a system of equations[END_REF] and the second argument uses Corollary 4. Theorem 6. Let 2 ≤ r ≤ n be two integers and b n,r be the best constant such that for all convex bodies A, B 1 , . . . , B r in

R n |A| r-1 V (B 1 , . . . , B r , A[n -r]) ≤ b n,r r i=1 V (B i , A[n -1]). Then, b n,r ≤ min k∈{1,...,r} 2 k(k-1) 2 n r-k (r-k)! .
Before proving this theorem, we introduce the following generalization of Fenchel's inequality (5). Lemma 7. Let 1 ≤ m ≤ n be two integers and A, B 1 , . . . , B m be convex bodies in R n . Then

|A|V (B 1 , . . . , B m , A[n -m]) ≤ 2 m-1 V (B 1 , . . . , B m-1 , A[n -m + 1])V (B m , A[n -1]) (15)
and

|A| m-1 V (B 1 , . . . , B m , A[n -m]) ≤ 2 m(m-1) 2 m i=1 V (B i , A[n -1]). (16)
Proof: We prove (15) by induction on m. The case m = 1 is trivial and the case m = 2 is Fenchel's inequality [START_REF] Bernstein | The number of roots of a system of equations[END_REF]. Let us assume that ( 15) is verified for m -1. From Fenchel's inequality,

V (B 1 , . . . , B m , A[n -m])V (B 1 , . . . , B m-2 , A[n -m + 2]) ≤ 2V (B 1 , . . . , B m-1 , A[n -m + 1])V (B 1 , . . . , B m-2 , B m , A[n -m + 1]). (17)
By the induction hypothesis, we have

|A|V (B 1 , . . . , B m-2 , B m , A[n -m + 1]) ≤ 2 m-2 V (B 1 , . . . , B m-2 , A[n -m + 2])V (B m , A[n -1]). ( 18
)
We end the proof by multiplying term by term [START_REF] Ruzsa | The Brunn-Minkowski inequality and nonconvex sets[END_REF] and [START_REF] Saroglou | Characterization of simplices via the Bézout inequality for mixed volumes[END_REF]. The proof of ( 16) follows from ( 15) by induction on m. ✷ Proof of theorem 6: First we remark that, for any k ∈ {1, . . . , r}, by taking in [START_REF] Fradelizi | On the volume of the Minkowski sum of zonoids[END_REF]

i 1 = • • • = i m = 1, m = r -k and for 1 ≤ j ≤ k, K j = B r-k+j , for k ≤ j ≤ n -|i|}, K j = A, we get that V (B 1 , . . . , B r , A[n -r])|A| r-k ≤ n r-k (r -k)! V (B r-k+1 , . . . , B r , A[n -k]) r-k i=1 V (B i , A[n -1]),
According to (16), we have

V (B r-k+1 , . . . , B r , A[n -k])|A| k-1 ≤ 2 k(k-1) 2 r i=r-k+1 V (B i , A[n -1]),
Hence the result follows. ✷ 4. Bézout inequality for Minkowski sums.

In [START_REF] Ruzsa | The Brunn-Minkowski inequality and nonconvex sets[END_REF], Ruzsa showed that for any compacts sets A, B 1 , . . . , B m ⊆ R n and any ǫ > 0, if |A| = 0, there exists a compact set A ′ ⊂ A such that

|A| m |A ′ + B 1 + • • • + B m | ≤ (1 + ǫ)|A ′ | m k=1 |A + B k |. ( 19
)
It was noticed in [START_REF] Fradelizi | Sumsets estimates in convex geometry[END_REF] that [START_REF] Schneider | Convex bodies: the Brunn-Minkowski theory[END_REF] gives

|A| m-1 |B 1 + • • • + B m | ≤ m k=1 |A + B k |. ( 20
)
Notice that our methods allows to give a new simple proof of [START_REF] Soprunov | Bézout inequality for mixed volumes[END_REF]. Indeed, if |i| = n, inequality [START_REF] Fradelizi | On the volume of the Minkowski sum of zonoids[END_REF] becomes

n! i 1 ! . . . i m ! |A| m-1 V (B 1 [i 1 ], . . . , B m [i m ]) ≤ m k=1 n i k V (B k [i k ], A[n -i k ]) (21)
and since

|A| m-1 |B 1 + • • • + B m | = i 1 +•••+im=n n! i 1 ! . . . i m ! |A| m-1 V (B 1 [i 1 ], . . . , B m [i m ]), (22)
m k=1 |A + B k | = n i 1 =1 • • • n im=1 m k=1 n i k V (B k [i k ], A[n -i k ]). (23)
According to [START_REF] Szusterman | A new condition towards the Soprunov-Zvavitch on Bézout-type inequalities[END_REF], each term of the right side of ( 22) is less than a corresponding term of the right side of ( 23). Hence, we get a new proof of [START_REF] Soprunov | Bézout inequality for mixed volumes[END_REF]. Another way of proving ( 20) is to first show that for all m ≥ 1 and any semi-definite positive symmetric matrices A, B 1 , . . . , B m ,

det(A) m-1 det(B 1 + • • • + B m ) ≤ m k=1 det(A + B k ),
by extending each term into mixed discriminants like in ( 22) and ( 23) and applying Theorem 1.

Then we use again the same optimal transport method.

Theorem 8. Let m, n ≥ 1, two integers and c n,m the best positive constant such that

|A| m-1 |A + B 1 + • • • + B m | ≤ c n,m m k=1 |A + B k |, (24)
for any convex bodies A, B 1 , . . . , B m ⊂ R n . Then (i)

c n,m ≤ 1 -x m 1 -mx m n < 2 n ,
where x m is the unique real root of

P r (x) = (1 -mx) m -(m -1) m-1 x m-1 (1 -x). (ii) c n,m ≥ e 2πn(e -1) g(n, m) -(n+ 1 2 ) ,
where

g(n, m) = 1 - 1 m -1 m -1 m m m-1
.

Furthermore, 4 3 ≤ g(n, m) < e e -1 Note that Theorem 8 does not follow from [START_REF] Schneider | Convex bodies: the Brunn-Minkowski theory[END_REF] and the sets in (24) are convex bodies.

Remark 9.

Here we give some simpler upper bounds of c n,m (i) for fixed n, (c n,m ) m≥2 is a increasing sequence with respect to m, 1 ≤ c n,2 ≤ c n,m ≤ c n,m+1 ≤ (c n,2 ) m and for q > 0, we have c n,m+q ≤ c n,m c n,q+1 . In fact, in (24), for m ≥ 3, if B m = {0}, we get

|A| m-2 |A + B 1 + • • • + B m-1 | ≤ c n,m m-1 k=1 |A + B k |, hence, c n,m-1 ≤ c n,m . If B k = {0}
for all 1 ≤ k ≤ m then we obtain equality in (24) with c n,m = 1, so c n,m ≥ 1. It is shown in [START_REF] Fradelizi | Sumsets estimates in convex geometry[END_REF] that 1 = c 2,2 ≤ c n,2 by using [START_REF] Bernstein | The number of roots of a system of equations[END_REF]. In (24), if we replace B m by

B m + • • • + B m+q , we get |A| m-1 |A + B 1 + • • • + B m+q | ≤ c n,m |A + B m + • • • + B m+q | m-1 k=1 |A + B k |. (25)
On the other hand, we have

|A| q |A + B m + • • • + B m+q | ≤ c n,q+1 m+q k=m |A + B k |. (26)
Thus, ( 25) and ( 26) allow to say that c n,m+q ≤ c n,m c n,q+1 , in particular, c n,m ≤ (c n,2 ) m-1 .

(ii) We also remark that a weaker bound for c n,m can be obtained by applying [START_REF] Soprunov | Bézout inequality for mixed volumes[END_REF] to

1 m A + B i instead of B i for i ∈ {1, . . . , m} and 1 -1 m A instead of A. This gives |A| m-1 |A + B 1 + • • • + B m | ≤ 1 + 1 m -1 m-1 n m k=1 |A + B k |, so c n,m ≤ 1 + 1 m-1 m-1 n ≤ e n .
(iii) We extend again as sums both sides of (24) like ( 22) and ( 23) and compare each term of

|A| m-1 |A + B 1 + • • • + B m |
= i 1 + • • • + i m ≤ n n! i 1 ! . . . i m !(n -|i|)! |A| m V (A[n -|i|], B 1 [i 1 ], . . . , B m [i m ]) ≤ d n,m m k=1 n i k V (B k [i k ], A[n -i k ]).
In [START_REF] Fradelizi | On the volume of the Minkowski sum of zonoids[END_REF], if

K 1 = • • • = K n-|i| = A, it becomes n! i 1 ! . . . i m !(n -|i|)! |A| m-1 V (A[n -|i|], B 1 [i 1 ], . . . , B m [i m ]) ≤ n |i| m k=1 n i k V (B k [i k ], A[n -i k ]), so any d n,m such that n |i| ≤ d n,m for all |i| ≤ n gives an upper bound for c n,m , hence c n,m ≤ max |i|≤n n |i| = n ⌊ n 2 ⌋ < 2 n .

Proof of Theorem 8 (i) (upper bound):

The following idea comes from [START_REF] Fradelizi | Sumsets estimates in convex geometry[END_REF] where the authors prove that c n,2 ≤ ϕ n (where ϕ is the golden ratio). Similarly to computations (iii) from Remark 9, we recall that after expending both sides of (24) and comparing term by term, one has c n,m ≤ d n,m where d n,m satisfies for all |i|

:= i 1 + • • • + i m ≤ n n! i 1 ! . . . i m !(n -|i|)! |A| m V (A[n -|i|], B 1 [i 1 ], . . . , B m [i m ]) ≤ d n,m m k=1 n i k V (B k [i k ], A[n -i k ]), thus |A| m V (A[n -|i|], B 1 [i 1 ], . . . , B m [i m ]) ≤ i 1 ! . . . i m !(n -|i|)! n! d n,m m k=1 n i k V (B k [i k ], A[n -i k ]).
For any j ∈ {1, . . . , m}, (13) can be rewritten as

V (B 1 [i 1 ], . . . , B m [i m ], A[n -|i|])|A| m-1 ≤ i 1 ! . . . i m !(n -i j )! (|i| -i j )!n! m k=1 n i k V (B k [i k ], A[n -i k ]).
Therefore, for a fixed j ∈ {1, . . . , m}, to obtain an upper bound on c n,m , it is enough to find d n,m such that for any |i| ≤ n

i 1 ! . . . i m !(n -i j )! (|i| -i j )!n! ≤ i 1 ! . . . i m !(n -|i|)! n! d n,
(1 -x j ) 1-x j (1 -|x|) 1-|x| (|x| -x j ) |x|-x j n .
For |x| ≤ n and j ∈ {1, . . . , m}, let

F j (x) = (1 -x j ) 1-x j (1 -|x|) 1-|x| (|x| -x j ) |x|-x j .
One has ∇ 2 log F j (x) = -1 1-|x| U U t where U j = |x|-x j 1-x j and U k = 1/U j for k = j, then, F j is log-concave and min log F j also. Note that min 1≤j≤m F j is symmetric with respect to each hyperplane

{x i = x k }, therefore max |x|≤1 min j∈{1,...,m} F j (x) is reached only if x 1 = • • • = x m .
As a result, the problem becomes a simple study of a function having one variable:

c n,m ≤ max x≤1/m (1 -x) 1-x (1 -mx) 1-mx ((m -1)x) (m-1)
x n and this maximum is reached at the unique x m ∈]0, 1/m[ which is the real root of P r (

x) = (1 -mx) m -(m -1) m-1 x m-1 (1 -x). After some simplifications, we get c n,m ≤ 1-xm 1-mxm n < 2 n . For m = 2, x 2 = 5- √ 5 2
and 1-x 2 1-2x 2 = ϕ (golden ratio), for m = 3 we get that c n,3 < (1.755) n . ✷

For the lower bound of c n,m , many results have been obtained in [START_REF] Alonso-Gutiérrez | Rogers-Shephard and local Loomis-Whitney type inequalities[END_REF][START_REF] Artstein-Avidan | Remarks about mixed discriminants and volumes[END_REF][START_REF] Fradelizi | Some inequalities about mixed volumes[END_REF][START_REF] Fradelizi | Sumsets estimates in convex geometry[END_REF][START_REF] Fradelizi | On the volume of the Minkowski sum of zonoids[END_REF][START_REF] Soprunov | Bézout inequality for mixed volumes[END_REF]. For the case m = 2 we use methods from [START_REF] Alonso-Gutiérrez | Rogers-Shephard and local Loomis-Whitney type inequalities[END_REF][START_REF] Fradelizi | On the volume of the Minkowski sum of zonoids[END_REF][START_REF] Fradelizi | Sumsets estimates in convex geometry[END_REF] and extend them to m ≥ 2. The idea is to represent volumes in (24) as volumes of a projection of A to subspaces of lower dimension and optimize the lower bound with respect to those dimensions. We remark that (24) is equivalent to

|A| m-1 |A + B 1 + • • • + B m | m k=1 |A + B k | ≤ c n,m .
So any choice of A, B 1 , . . . , B m gives a lower bound of c n,m . Let Γ 2 be the set of vector subspaces

{E 1 , E 2 } such that dim(E 1 ) = i, dim(E 2 ) = j, 0 < i, j ≤ n, i + j > n. It is obtained in [2] that for any A convex body in R n , c n,2 ≥ max A,Γ 2 |A| n |P E 1 ∩E 2 A| i+j-n |P E 1 A| i |P E 2 A| j = i i+j-n j i+j-n n i+j-n (27)
where A is a convex body in R n and P E A is the projection of A onto the subspace E. The equality case in ( 27) is reached for a sharp choice of E 1 , E 2 and A. The authors of [START_REF] Fradelizi | Sumsets estimates in convex geometry[END_REF] conclude that

c n,2 ≥ max 0<i,j≤n i+j>n i i+j-n j i+j-n n i+j-n ≈ 2 √ πn 4 3 n .
By following the same idea, we get a lower bound for c n,m .

Theorem 10. Let α 1 , . . . , α m ∈ {1, . . . , n} be integers and d = |α| -(m -1)n > 0. Let A ⊂ R n be a convex body and Γ m the set of collections of vector subspaces {E 1 , . . . , E m } such that for i = 1, . . . , m, dim(E i ) = α i and

E ⊥ i ⊂ j∈[m] {i} E j }. Then c n,m ≥ max Γm,A convex body |A| n |P m i=1 E i A| d m i=1 |P E i A| α i . (28)
Proof: Let (e 1 , . . . , e n ) be the canonical basis of R n . For i = 1, . . . , m, let

B i = k∈[n] K i [0, e k ],
where

K i = {α 1 + • • • + α i-1 + k|1 ≤ k ≤ α i }.
Thus, for any i = 1, . . . , m and any t > 0, one has

|A + tB i | n ∼ t→∞ t n-α i |P E i A| α i and |A + tB 1 + • • • + tB m | n ∼ t→∞ t nm-|α| P m i=1 B i A d .
Let t to infinity, then we get

|A| m-1 n |P m i=1 E i A| d ≤ c n,m m i=1 |P E i A| α i .
The desired result follows. ✷

  by the term of the same index in m k=1 |A + B k |. It follows that c n,m ≤ d n,m where d n,m satisfies for all |i| :

Corollary 11 . 1 , 2 ( 1 - 2 ( 1 - 2 .For m = 2 , c n,2 ≥ 2 √ πn 4 3 n ( 1 +

 111212122231 Let α 1 , . . . , α m ∈ [1, n] be integers and d = |α| -(m -1)n > 0. Then c n,m ≥ max Let B ∞ (d) = n k=n-d+1 [-e k , e k ]. For i = 1, . . . , m, letJ i = [n -α 1 + . . . + n -α i-1 + 1 ; n -α 1 + . . . + n -α i ], B 1 (i) = conv J i (±e k ), and E i = vect{e k , k ∈ [n] J i }. If A = conv { m i=1 B 1 (i), B ∞ (d)}, we get that |A| = m i=1 |B 1 (i)| × |B ∞ (d)|/ n d ; |P m i=1 E i A| = |B ∞ (d)|and for i = 1, . . . , m|P E i A| = k∈m {i} |B 1 (k)| × |B ∞ (d)|/ α i d .Thus simplifying the volumes we get, for all α satisfying the given hypothesis. Hence the result follows. ✷ Proof of Theorem 8 (ii) (lower bound): For the lower bound, Corollary 11 allows to say thatc n,m ≥ max α>0 d=|α|-(m-1-d) α i -d e o(1) .For i = 1, . . . , m, one assume that α i = nx i and d = ny, then y= |x| -(m --d) α i -d = max x f (x) n , where f (x 1 , . . . , x m ) = (1 -y) (1-y)(m-1) -y) x i -y .By studying this function, we easily find that max f = x m y > 1 wherex = (m -1) m -m -1 m m-1 -1 and y = m(x -1) + 1. It follows that c n,m ≥ max x m y y) m-1 (x -y) m (1 + o(1sequence with respect to m while (1 -y) m-1 (x -y) m = y) m-1 (x -y) m < √6(e e -1 ) n+1/2 , o(1)) which was found in[START_REF] Fradelizi | Sumsets estimates in convex geometry[END_REF]. For m = 3, we find c(n,

  n as claimed in Remark 9. For any j ∈ {1, . . . , m}, let i j = x j n with x j ≥ 0 and |x| :=x 1 + • • • + x m ≤ 1.According to Stirling formula, for any integers k ≤ n, one has n k ≤

						m ,
	then, max |i|≤n	n-i j |i|-i j	≤ d n,m and it follow that
				c n,m ≤ max |i|≤n	min j∈{1,...,m}	n -i j |i| -i j	.
	Note that max |i|≤n	min j∈{1,...,m}	n-i j |i|-i j	≤ max |i|≤n	n |i| < 2 n n (n-k) n-k k k , therefore:
			c n,m ≤ max |x|≤1	min j∈{1,...,m}

Acknowledgement: We are grateful to Artem Zvavitch and Matthieu Fradelizi for introducing us to the question and their help. We also thank Pascal Dingoyan, Alfredo Hubard and Maud Szusterman for interesting discussions about this work.