On the volume of the John-L¨ownerellipsoid Grigory Ivanov1 Abstract. We find an optimal upper bound on the volume of the John ellipsoid of a 푘- dimensional section of the 푛-dimensional cube, and an optimal lower bound on the volume of the L¨ownerellipsoid of a projection of the 푛-dimensional cross-polytope onto a 푘-dimensional subspace. We use these results to give a new proof of Ball’s upper bound on the volume of a 푘- dimensional section of the hypercube, and of Barthe’s lower bound on the volume of a projection of the 푛-dimensional cross-polytope onto a 푘-dimensional subspace. We settle equality cases in these inequalities. Also, we describe all possible vectors in R푛, whose coordinates are the squared lengths of a projection of the standard basis in R푛 onto a 푘-dimensional subspace. Subject Classification (2010): 15A45, 52A38, 49Q20, 52A40 Keywords: John ellipsoid, L¨ownerellipsoid, section of the hypercube, projection of the cross-polytope, unit decomposition.

1. Introduction In [1], Fritz John proved that each convex body in R푘 contains a unique ellipsoid of maximal volume. John characterized all convex bodies 퐾 such that the ellipsoid of maximal volume in 퐾 is the Euclidean unit ball, ℰ푘. Theorem 1.1 (F. John). The Euclidean ball is the ellipsoid of maximal volume contained in 푘 푛 a convex body 퐾 ∈ R iff ℰ푘 ⊂ 퐾 and, for some 푛 > 푘, there are Euclidean unit vectors (푢푖)1 , 푛 on the boundary of 퐾, and positive numbers (푐푖)1 for which 푛 ∑︀ (1) 푐푖푢푖 = 0; 1 푛 ∑︀ 푘 (2) 푐푖푢푖 ⊗ 푢푖 = 퐼푘, the identity on R . 1 Keith Ball in [5] added the converse part to this theorem. 푛 푘 푛 Theorem 1.2 (K. Ball). Let (푢푖)1 be a sequence of unit vectors in 푅 and (푐푖)1 be a sequence of positive numbers satisfying (1) and (2). 푘 Then the set 퐾 = {푥 ∈ R |⟨푥, 푢푖⟩ 6 1, 푖 ∈ 1, 푛} contains a unique ellipsoid of maximal volume, which is the Euclidean unit ball. 푛 푛 The use of vectors (푢푖)1 and positive numbers (푐푖)1 satisfying (1) and (2) appears to be extremely powerful in a range of problems in , including (see [6] ): tight bounds on the volume ratio and on the outer volume ratio for centrally-symmetric convex bodies, and optimal upper bounds on the volume of a 푘-dimensional section of the 푛-cube. 푛 푘 arXiv:1707.04442v2 [math.FA] 25 Sep 2017 In this short paper we study a simple alternative description of the vectors (푢푖)1 ⊂ R and 푛 positive numbers (푐푖)1 satisfying (2). 푛 Definition 1.3. We will say that some vectors (푣푖)1 ⊂ 퐻 give a unit decomposition in a 푘-dimensional vector space 퐻 if 푛 ∑︁ (1.1) 푣푖 ⊗ 푣푖 = 퐼퐻 , 1 1 DCG, FSB, Ecole Polytechnique F´ed´eralede Lausanne, Route Cantonale, 1015 Lausanne, Switzerland. Department of Higher Mathematics, Moscow Institute of Physics and Technology, Institutskii pereulok 9, Dol- goprudny, Moscow region, 141700, Russia. [email protected] Research partially supported by Swiss National Science Foundation grants 200020-165977 and 200021-162884 Supported by Russian Foundation for Basic Research, project 16-01-00259. 1 2 where 퐼퐻 is the identity operator in 퐻. 푛 푘 푘 Clearly, non-zero vectors (푣푖)1 ⊂ R give a unit decomposition in R iff the vectors (︁ )︁푛 푣푖 푘 2 푛 |푣 | ⊂ R and positive numbers (|푣푖| )1 satisfy (2). 푖 1 In Lemma 2.5, the set of all possible vectors of positive reals (푐1, ··· , 푐푛), which we can get from condition (2). That this result may be interesting for finding optimal bounds in different geometric inequalities, including the Brascamb–Lieb inequality. Using a geometric approach, we will give in Theorem 3.3 an optimal upper bound on the volume of the John ellipsoid of a 푘-dimensional section of the 푛-dimensional cube, and derive an optimal lower bound on the volume of the L¨ownerellipsoid of a projection of the 푛-dimensional cross-polytope onto a 푘-dimensional subspace. In Section4 we give a new proof for Ball’s upper bound on the volume of a 푘-dimensional section of the hypercube (see [8]) and for Barthe’s lower bound on the volume of a projection of the 푛-dimensional cross-polytope onto a 푘-dimensional subspace (see [3]). Moreover, we settle equality cases in these inequalities. We use ⟨푝, 푥⟩ to denote the value of a linear functional 푝 at a vector 푥. For a convex body 퐾 ⊂ R푛 we denote by 퐾∘ and vol 퐾 the polar body and the volume of 퐾, respectively. We use 푛 푛 ♦ ,  to denote the 푛-dimensional cross-polytope and cube, respectively. For a convex body 푛 푛 퐾 ⊂ R and a 푘-dimensional subspace 퐻푘 of R we denote by 퐾 ∩ 퐻푘 and 퐾|퐻푘 the section of 퐾 by 퐻푘 and the orthogonal projection of 퐾 onto 퐻푘, respectively. 2. Properties of a unit decomposition In this section, using a simple linear algebra observation we introduce a description of sets of 푛 푘 푘 푘 푛 vectors (푣푖)1 ⊂ R which give a unit decomposition in R . Here we understand R ⊂ R as the subspace of vectors, whose last 푛 − 푘 coordinates are zero. For convenience, we will consider 푛 푘 푛 (푣푖)1 ⊂ R ⊂ R to be 푘-dimensional vectors. Lemma 2.1. The following assertions are equivalent: 푛 푘 푘 (1) vectors (푣푖)1 ⊂ R give a unit decomposition in R ; 푛 (2) there exists an orthonormal basis {푓1, ··· , 푓푛} in R such that 푣푖 is the orthogonal 푘 projection of 푓푖 onto R , for any 푖 ∈ 1, 푛; 푘 푘 (3) Lin{푣1, ··· , 푣푛} = R and the Gram matrix Γ of vectors {푣1, ··· , 푣푛} ⊂ R is the matrix of a projection operator from R푛 onto the linear hull of the rows the matrix 퐴 = [푣1, ··· , 푣푛]. (4) the 푘 × 푛 matrix 퐴 = [푣1, ··· , 푣푛] is a sub-matrix of an orthogonal matrix of order 푛. Proof. 1) (4) ⇒ (3). Since 퐴 is a sub-matrix of an orthogonal matrix, we have that rk 퐴 = 푘. Therefore, 푘 Lin{푣1, ··· , 푣푛} = R . Let Γ be the Gram matrix of vectors {푣1, ··· , 푣푛} and 푃 be the matrix of a projection 푛 operator from R onto the linear hull of the rows of the matrix 퐴 = [푣1, ··· , 푣푛]. Since the rows of 퐴 form an orthonormal basis of their linear hull 퐻푘, we can identify 푃 푒푖 and 푣푖 in this basis of 퐻푘. Therefore, 2 푃푖푗 = ⟨푃 푒푖, 푒푗⟩ = ⟨푃 푒푖, 푒푗⟩ = ⟨푃 푒푖, 푃 푒푗⟩ = ⟨푣푖, 푣푗⟩ = Γ푖푗.

2) (3) ⇒ (2). Let the Gram matrix Γ of the vectors {푣1, ··· , 푣푛} be the matrix of a projection operator onto 퐻푘. By the last identity, we have that ⟨Γ푒푖, Γ푒푗⟩ = ⟨푣푖, 푣푗⟩. But if two sets 푆1 and 푆2 of vectors have the same Gram matrix, then there exists an orthogonal transformation of the space that maps vectors of 푆1 to 푆2. Indeed, each step in the Gram-Schmidt process for both 3 systems are identical, that means that any orthogonal transformation which maps the Gram- Schmidt orthonormalization of 푆1 to the Gram-Schmidt orthonormalization of 푆2 maps 푆1 to 푛 푆2. Therefore, with a proper choice of the orthonormal basis in R , we can identify vectors 푣푖 푘 and 푃 푒푖, for 푖 ∈ 1, 푛, ans subspaces 퐻푘 and R = Lin{푣1, ··· , 푣푛}. 3)(2) ⇒ (1) 푛 푘 푘 Let 푃 be the projection from R onto R . Let 푣푖 = 푃 푓푖. For an arbitrary vector 푥 ∈ R we have 푃 푥 = 푥 and therefore (︃ 푛 )︃ 푛 푛 푛 푛 ∑︁ ∑︁ ∑︁ ∑︁ ∑︁ 푣푖 ⊗ 푣푖 푥 = ⟨푣푖, 푥⟩푣푖 = ⟨푃 푓푖, 푥⟩푣푖 = ⟨푓푖, 푃 푥⟩푣푖 = 푃 ( ⟨푓푖, 푥⟩푓푖) = 푃 푥 = 푥. 1 1 1 1 1

4)(1) ⇒ (4) 푛 ∑︀ For 푖 ∈ 1, 푘 we have 푒푗 = ⟨푣푖, 푒푗⟩푣푖. Therefore, 1 푛 푛 2 ∑︁ 2 ∑︁ 2 1 = |푒푗| = ⟨푣푖, 푒푗⟩ = 푣푖[푗] , 1 1 a2nd 푛 푛 ∑︁ ∑︁ 0 = ⟨푒푗, 푒푘⟩ = ⟨푣푖, 푒푗⟩⟨푣푖, 푒푘⟩ = 푣푖[푗]푣푖[푘], 1 1 where 푣푖[푗] is the 푗’th coordinate of the vector 푣푖 in the given basis. That is, the rows of the 푘 × 푛 matrix [푣1, ··· , 푣푛] form an orthonormal system of 푘 vectors in R푛. 

As a direct consequence of Lemma 2.1 we get

푛 푘 푛 Corollary 2.2. Let (푢푖)1 be a sequence of unit vectors in 푅 and (푐푖)1 be a sequence of positive 푘 numbers satisfying (2). Then the set 퐾 = {푥 ∈ R ||⟨푥, 푢푖⟩| 6 1, 푖 ∈ 1, 푛} is an affine image of 푛 a 푘-dimensional section of  . 푘 Definition 2.3. We will say that a vector 퐶 = (푐1, ··· , 푐푛) is realizable in R if there exist 푛 푘 2 vectors (푣푖)1 which give a unit decomposition in R such that 푐푖 = |푣푖| , 푖 ∈ 1, 푛. Now we are going to describe all possible realizable vectors in R푘. For this purpose, we need to use the following standard notation.

Definition 2.4. Let 푎 and 푏 be non-negative vectors in R푛. The vector 푎 majorizes the vector 푏, which we denote by 푎 ≻ 푏, if the sum of the 푘 largest entries of 푎 is at least the sum of the 푘 largest entries of 푏, for every 푘 ∈ 1, 푛, and the sums of all entries of 푎 and 푏 are equal.

Lemma 2.5. A vector (푐1, ··· , 푐푛) is realizable iff

(1, ··· , 1, 0, ··· , 0) ≻ (푐1, ··· , 푐푛). ⏟ ⏞ ⏟ ⏞ 푘 푛−푘 Proof. Let (푐1, ··· , 푐푛) be a realizable vector. By definition and by Lemma 2.1, there are vectors 푛 푘 푘 (푣푖)1 ⊂ R that give a unit decomposition in R such that the diagonal entries of their Gram 푛 푛 matrix Γ are (푐푖)1 , and Γ is the matrix of a projection operator from R onto some 푘-dimensional subspace 퐻푘. 푛 So the vector (푐1, ··· , 푐푛) is realizable iff there exists a projection operator from R onto some 푘-dimensional subspace with (푐1, ··· , 푐푛) on the main diagonal. Applying Horn’s theorem [7], which asserts that a vector (푐1, ··· , 푐푛) can be the main diagonal of a Hermitian matrix 4 with a vector of eigenvalues (휆1, ··· , 휆푛) iff 휆 ≻ 푐, to the vector (1, ··· , 1, 0, ··· , 0) we complete ⏟ ⏞ ⏟ ⏞ 푘 푛−푘 the proof. 

3. Estimation for the volume of the Lowner-John¨ ellipsoid Before stating the next result, we recall that the John ellipsoid of a convex body 퐾 is the ellipsoid of maximal volume contained in 퐾, and the L¨ownerellipsoid of a convex body 퐾 is the 퐻푘 ellipsoid of minimal volume containing 퐾. We use ℰ퐻푘 and ℰ to denote the L¨ownerellipsoid 푛 푛 of ♦ |퐻푘 and the John ellipsoid of  ∩ 퐻푘, respectively. 푛 푘 푘 Lemma 3.1. Suppose vectors (푣푖)1 ⊂ R give a unit decomposition in R . Then for any 푛 ellipsoid ℰ with the center in the origin that covers all vectors (푣푖)1 , we have 푘 (︂푘 )︂ 2 (3.1) vol ℰ vol ℰ , > 푛 푘 푘 where ℰ푘 is the unit ball in R . The bound is tight. The inequality becomes an equality iff 2 푘 |푣푖| = 푛 , for all 푖 ∈ 1, 푛. Proof. 푘 푘 For a positive-definite operator 퐴 on R , we know that the volume of the ellipsoid {푥 ∈ R |⟨퐴푥, 푥⟩ 6 1} is √vol ℰ푘 . To prove our lemma, it is enough to show that for any positive-definite operator 퐴 det 퐴 (︀ 푛 )︀푘 such that ⟨퐴푣푖, 푣푖⟩ 6 1 for 푖 ∈ 1, 푛 we have det 퐴 6 푘 . 푘 Fix a positive-definite operator 퐴 on R such that ⟨퐴푣푖, 푣푖⟩ 6 1 for 푖 ∈ 1, 푛. We can choose an 푘 ′ orthonormal basis in R such that 퐴 = diag{휆1, ··· , 휆푘} in this basis. Let 푣푖 be the coordinate vector of 푣푖 in the new basis for each 푖 ∈ 1, 푛. We can rewrite the inequality ⟨퐴푣푖, 푣푖⟩ 6 1 in the following form: 푘 ∑︁ ′ 2 (3.2) 휆푗푣푖[푗] 6 1, 1 ′ ′ where 푣푖[푗] is the 푗’th coordinate of the vector 푣푖 in the given basis. 푛 ∑︀ ′ 2 Summing up the inequalities (3.2) for all 푖 ∈ 1, 푛 and using the observation that 푣푖[푗] = 1 1 (see Lemma 2.1), we get 푘 ∑︁ 휆푖 6 푛. 1 Applying the AM-GM inequality, we get ⎛ 푘 ⎞푘 ∑︀ 푘 휆 푖 푘 ∏︁ ⎜ 1 ⎟ (︁푛)︁ det 퐴 = 휆푖 ⎜ ⎟ . 6 푘 6 푘 1 ⎝ ⎠

(︀ 푘 푘 )︀ According to Lemma 2.5, the vector 푛 , ··· , 푛 is realizable. It is clear that in this case inequality (3.1) becomes an equality. Moreover, by properties of the AM-GM inequality, we 푛 have an equality in (3.1) iff 휆푖 = 푘 , for all 푖 ∈ 1, 푛. By the inequality (3.2), there is an equality 2 푘 in (3.1) iff |푣푖| = 푛 , for all 푖 ∈ 1, 푛. 

푛 푛 Fix a 푘-dimensional subspace 퐻푘 in R . Let 푃 : R → 퐻푘 be the projection onto 퐻푘. Since 푛 ♦ |퐻푘 is the absolute of the vectors 푣푖 = 푃 푒푖 for 푖 ∈ 1, 푛 that give us a unit 5

푛 decomposition in 퐻푘, Lemma 3.1 implies that the volume of the L¨ownerellipsoid for ♦ |퐻푘 is 푘 (︀ 푘 )︀ 2 at least 푛 vol ℰ푘. To settle the reverse case, we need to recall the following simple arguments. 푛 푛 * 푛 For a given 푘-dimensional subspace 퐻푘 in R , we can consider the space 퐻푘 ⊂ (R ) = R itself to be the dual space for 퐻푘. Indeed, 퐻푘 is a 푘-dimensional space consisting of all linear functionals on 퐻푘 with the proper linear structure, and the restriction of the Euclidean norm 푛 in R onto 퐻푘 generates the operator norm on 퐻푘. For the sake of completeness we give a proof of the following well-known fact.

푛 * Lemma 3.2. Let 퐻푘 be a 푘-dimensional subspace of R . Assume the dual space 퐻푘 for 퐻푘 is 푛 퐻푘 itself. For a convex body 퐾 ∈ R containing the origin in the interior, we have

∘ ∘ (퐾 ∩ 퐻푘) = 퐾 |퐻푘,

* where we understand 퐾 ∩ 퐻푘 as a subset of 퐻푘, and its polar set as a subset of 퐻푘 = 퐻푘. Proof. ⊥ 푛 We use 퐻푘 to denote the orthogonal complement of 퐻푘 in R . ∘ ∘ ∘ 1) Let us show that (퐾 ∩ 퐻푘) ⊃ 퐾 |퐻푘. Fix a functional 푝 ∈ 퐾 |퐻푘. Since 푝 belongs to the ∘ ⊥ ⊥ ⊥ ∘ projection of 퐾 , there is a functional 푝 ⊂ 퐻푘 such that 푝 + 푝 ∈ 퐾 . By definition of the ⊥ polar body, we have ⟨푝 + 푝 , 푥⟩ 6 1 for any 푥 ∈ 퐾. In particular, for any 푥 ∈ 퐾 ∩ 퐻푘, we have

⊥ ⊥ 1 > ⟨푝 + 푝 , 푥⟩ = ⟨푝, 푥⟩ + ⟨푝 , 푥⟩ = ⟨푝, 푥⟩.

∘ This means that 푝 ∈ (퐾 ∩ 퐻푘) . ∘ ∘ 2) Let us show that (퐾∩퐻푘) ⊂ 퐾 |퐻푘. Suppose for a contradiction that there is a functional ∘ ∘ 푝 ∈ (퐾 ∩ 퐻푘) such that 푝∈ / 퐾 |퐻푘. By the hyperplane separation theorem, there exists a vector 푦 ∈ 퐻푘 such that (3.3) ⟨푝, 푦⟩ > 1 and

∘ (3.4) ⟨푞, 푦⟩ 6 1 for all 푞 ∈ 퐾 |퐻푘.

⊥ ⊥ ⊥ Clearly, ⟨푦, 푞 ⟩ = 0 for any 푞 ∈ 퐻푘 . Combining this and the inequality (3.4), we get

⟨푞, 푦⟩ 6 1 for all 푞 ∈ 퐾∘. By the definition of the polar set, we obtain 푦 ∈ (퐾∘)∘ = 퐾. So 푦 ∈ 퐾 and 푦 ∈ 퐻푘, therefore 푦 ∈ 퐾 ∩ 퐻푘. This contradicts the inequality (3.3). 

For a given convex centrally-symmetric body 퐾 with the center at the origin, by symmetry and duality arguments, we have that the polar ellipsoid of the John ellipsoid of 퐾 is the L¨owner ellipsoid of 퐾∘. Summarizing the arguments of section 3, we obtain.

Theorem 3.3. For any 1 6 푘 6 푛 we have

푘 (︂ )︂ 2 퐻푘 푘 vol ℰ퐻푘 푘 vol ℰ (︁푛)︁ 2 > and 6 . vol ℰ푘 푛 vol ℰ푘 푘

The bounds are sharp. That is, there exists a subspace 퐻푘 such that the two inequalities are simultaneously hold as equalities. 6

푛 푛 4. Bounds on the volume of a section of  and a projection of ♦ K. Ball, in his fundamental paper [8], proved the following inequality 푛 푘 vol( ∩ 퐻푘) (︁푛)︁ 2 (4.1) 6 . vol 푘 푘 F. Barthe in [3] proved the dual inequality

푘 푛 (︂ )︂ 2 vol(♦ |퐻푘) 푘 (4.2) > . vol ♦푘 푛

One can see that both inequalities become equalities when 푘|푛 and 퐻푘 is determined by the system of linear equations 푛 (4.3) 푥 푛 푗+푖 = 푥 푛 푗+푖 , where 푗 ∈ 0, 푘 − 1 and 1 푖1, 푖2 . 푘 1 푘 2 6 6 푘 Using Theorem 3.3, we are going to give another proof of the inequalities (4.1) and (4.2), and settle the equality case.

Theorem 4.1. For any 푘-dimensional subspace of R푛, we have 푘 푛 푘 푛 (︂ )︂ 2 vol( ∩ 퐻푘) (︁푛)︁ 2 vol(♦ |퐻푘) 푘 6 and > . vol 푘 푘 vol ♦푘 푛 The bounds are optimal iff 푘|푛. Proof. Using the Brascamb–Lieb inequality, K. Ball [8] proved that among all 푘-dimensional convex 1 (︀ vol 퐾 )︀ 푛 centrally-symmetric bodies, the 푘-cube has the greatest volume ratio (i.e., vol ℰ , where ℰ is the John ellipsoid of 퐾). This means that

푛 퐻푘 vol( ∩ 퐻푘) vol ℰ (4.4) 푘 6 . vol  vol ℰ푘 1 (︀ vol ℰ )︀ 푛 The dual case for the outer volume ratio (i.e. vol 퐾 , where ℰ is the L¨ownerellipsoid of 푘 퐾) was resolved using Barthe’s reverse Brascamb–Lieb inequality [6]. It was shown that ♦ has the biggest outer volume ratio among all 푘-dimensional convex centrally-symmetric bodies. Therefore 푛 vol(♦ |퐻푘) vol ℰ퐻푘 (4.5) 푘 > . vol ♦ vol ℰ푘 Combining (4.4) and (4.5) with the inequalities from the assertion of Theorem 3.3, we obtain (4.1) and (4.2). We now prove that the bounds are optimal only if 푘|푛. In [3] Proposition 10, Barthe proved that whenever the volume ratio for a convex centrally- 푘 푘 symmetric body 퐾 ⊂ R equals the volume ratio for  , then 퐾 is an affine 푘-dimensional cube (or parallelotope). Also, he proved that if a centrally-symmetric convex body 퐾 ⊂ R푘 has the extremal inner volume ratio, then 퐾 is an affine cross-polytope. These arguments imply 푛 푛 that  ∩ 퐻푘 is an affine cube and ♦ |퐻푘 is an affine cross-polytope in the equality cases for the inequalities (4.1) and (4.2), respectively. 푘 푘 Using the fact that  is the polar set of ♦ and employing Lemma 3.2, we obtain that for any given subspace 퐻푘 equality holds in (4.1) if and only if equality holds in (4.2). Hence, it is enough to settle equality only for the inequality (4.2). 푛 Suppose for a given 퐻푘 we have equality in (4.2). Then ♦ |퐻푘 is an affine 푘-dimensional 푛 cross-polytope. Let 푃 be the projection from R onto 퐻푘, and 푣푖 = 푃 푒푖. It is easy to see that 푛 each vertex of ♦ |퐻푘 is identical to at least one of the vectors 푣푖, where 푖 ∈ 1, 푛. The proof of Lemma 3.1 yields that all lengths |푣푖|, for 푖 ∈ 1, 푛, are the same. From this and the triangle 7 inequality, we conclude that all vectors 푣푖, 푖 ∈ 1, 푛, are vertices of the affine cross-polytope 푛 푛 ♦ |퐻푘. So, each vertex of ♦ |퐻푘 is identical to some 푣푖, and conversely, each 푣푖 is a vertex of 푛 ♦ |퐻푘. Denote by ℓ푖, 푖 ∈ 1, 푘, lines in 퐻푘 that pass through vertices of the affine cross-polytope 푛 ♦ |퐻푘. We showed that for any 푖 ∈ 1, 푛 there exists 푗 ∈ 1, 푘 such that 푣푖 ∈ ℓ푗. Hence, there 푛 푘 ∑︀ ∑︀ exist vectors 푑푗 ∈ ℓ푗, 푗 ∈ 1, 푘, such that 퐼퐻푘 = 푣푖 ⊗ 푣푖 = 푑푗 ⊗ 푑푗. This means that the 1 1 vectors 푑푗, 푗 ∈ 1, 푘, give us a unit decomposition in 퐻푘. By the assertion4 of Lemma 2.1, we have that the vectors 푑푗, 푗 ∈ 1, 푘, form an orthonormal basis in 퐻푘. Therefore, all 푘 sums ∑︀ 2 |푣푖| , 푗 ∈ 1, 푘, equals 1. As mentioned above, all lengths |푣푖|, for 푖 ∈ 1, 푛, are the same. 푣푖∈ℓ푗

Consequently, the same number of vectors 푣푖, 푖 ∈ 1, 푛, lies on each line ℓ푗, 푗 ∈ 1, 푘. That is, 푘|푛.  Remark 4.2. Up to coordinate permutation and up to change of the sign of coordinates, Theorem 4.1 implies that equality in (4.1) and (4.2) is attained when 퐻푘 is determined by (4.3). We should note that Ball’s and Barthe’s proofs of the inequalities (4.1) and (4.2) used the same arguments as the proofs of (4.4) and (4.5). However, we believe that it may be of interest how our result reveals the connection between Theorem 4.1 and the volume of the L¨ownerand the John ellipsoid. We conjecture the following. Conjecture 4.3. 푛 vol(♦ |퐻푘) 푘−푛 (4.6) > 2 2 . vol ♦푘 The bound is optimal when 2푘 > 푛. This is the dual statement for another Ball’s upper bound on the volume of a 푘-dimensional 푛 section of  : 푛 vol( ∩ 퐻푘) 푛−푘 (4.7) 6 2 2 . vol 푘 We note that inequalities (4.2) and (4.6) follow from the well-known Mahler conjecture and inequalities (4.1) and (4.7), respectively. Acknowledgements. I am grateful for Marton Naszodi for useful remarks and help with the text. I am especially grateful for Roma Karasev for fruitful conversations concerned the topic of this paper.

References [1] F. John. Extremum problems with inequalities as subsidiary conditions. In Traces and emergence of nonlinear programming, pages 197–215. Springer, 2014. [2] C. Zong. The Cube-A Window to Convex and Discrete , volume 168. Cambridge University Press, 2006. [3] F. Barthe. On a reverse form of the brascamp-lieb inequality. Inventiones mathematicae, 134(2):335–361, 1998. [4] S. Brazitikos, A. Giannopoulos, P. Valettas, and B.-H. Vritsiou. Geometry of isotropic convex bodies, volume 196. American Mathematical Society Providence, 2014. [5] K. Ball. Ellipsoids of maximal volume in convex bodies. Geometriae Dedicata, 41(2):241–250, 1992. [6] K. Ball. and functional analysis. Handbook of the geometry of Banach spaces, 1:161–194, 2001. [7] A. Horn. Doubly stochastic matrices and the diagonal of a rotation matrix. American Journal of Mathemat- ics, 76(3):620–630, 1954. [8] K. Ball. Volumes of sections of cubes and related problems. Geometric aspects of functional analysis, pages 251–260, 1989.