On Discrete LOG-Brunn--Minkowski Type Inequalities

Mahler's conjecture asks whether the cube is a minimizer for the volume product of a body and its polar in the class of symmetric convex bodies in R^n. The corresponding inequality to the conjecture is sometimes called the the reverse Blaschke-Santalo inequality. The conjecture is known in dimension two and in several special cases. In the class of unconditional convex bodies, Saint Raymond confirmed the conjecture, and Meyer and Reisner, independently, characterized the equality case. In this paper we present a stability version of these results and also show that any symmetric convex body, which is sufficiently close to an unconditional body, satisfies the the reverse Blaschke-Santalo inequality.

In 1999, Dar conjectured that there is a stronger version of the celebrated Brunn-Minkowski inequality. However, as pointed out by Campi, Gardner, and Gronchi in 2011, this problem seems to be open even for planar $o$-symmetric convex bodies. In this paper, we give a positive answer to Dar’s conjecture for all planar convex bodies. We also give the equality condition of this stronger inequality. For planar $o$-symmetric convex bodies, the log–Brunn–Minkowski inequality was established by Boroczky, Lutwak, Yang, and Zhang in 2012. It is stronger than the classical Brunn–Minkowski inequality, for planar $o$-symmetric convex bodies. Gaoyong Zhang asked if there is a general version of this inequality. Fortunately, the solution of Dar’s conjecture, especially, the definition of “dilation position”, inspires us to obtain a general version of the log–Brunn–Minkowski inequality. As expected, this inequality implies the classical Brunn–Minkowski inequality for all planar convex bodies.

High dimensional random sections of isotropic convex bodies

The cosine representation of the support function of a centrally symmetric convex body plays a fundamental role in integral geometry. In this article, one new so-called flag representation for the support function of an origin symmetric n-dimensional convex body in terms of surface curvature functions of the convex body is found. Using the representation, we propose a sufficient condition for an origin symmetric n-dimensional convex body to be a zonoid. The condition has a local equatorial description.

We prove a number of statements concerning lattice packings of mirror symmetric or centrally symmetric convex bodies. This enables one to establish the existence of sufficiently dense lattice packings of any three-dimensional convex body of such type. The main result states that each three-dimensional, mirror symmetric, convex body admits a lattice packing with density at least 8/27. Furthermore, two basis vectors of the lattice generating the packing can be chosen parallel to the plane of symmetry of the body. The best result for centrally symmetric bodies was obtained by Edwin Smith (2005): Each three-dimensional, centrally symmetric, convex body admits a lattice packing with density greater than 0.53835. In the present paper, it is only proved that each such body admits a lattice packing with density $$ \left(\sqrt{3}+\sqrt[4]{3/4}+1/2\right)/6>0.527 $$ . Bibliography: 5 titles.

We introduce both the notions of tensor product of convex bodies that contain zero in the interior, and of tensor product of $0$-symmetric convex bodies in Euclidean spaces. We prove that there is a bijection between tensor products of $0$-symmetric convex bodies and tensor norms on finite dimensional spaces. This bijection preserves duality, injectivity and projectivity. We obtain a formulation of Grothendieck`s Theorem for $0$-symmetric convex bodies and use it to give a geometric representation (up to the $K_G$-constant) of the Hilbertian tensor product. We see that the property of having enough symmetries is preserved by these tensor products, and exhibit relations with the L\"owner and the John ellipsoids.

In this paper we study geometry of compact, not necessarily centrally symmetric, convex bodies in R. Over the years, local theory of Banach spaces developed many sophisticated methods to study centrally symmetric convex bodies; and already some time ago it became clear that many results, if valid for arbitrary convex bodies, may be of interest in other areas of mathematics. In recent years many results on non-centrally symmetric convex bodies were proved and a number of papers have been written (see e.g., [1], [8], [12], [18], [27], [28] among others). The present paper concentrates on random aspects of compact convex bodies and investigates some invariants fundamental in the local theory of Banach spaces, restricted to random sections and projections of such bodies. It turns out that, loosely speaking, such random operations kill the effect of non-symmetry in the sense that resulting estimates are very close to their centrally symmetric counterparts (this is despite the fact that random sections might be still far from being symmetric (see Section 5 below)). At the same time these estimates might be in a very essential way better than for general bodies. We are mostly interested in two directions. One is connected with so-called MM∗estimate, and related inequalities. For a centrally symmetric convex body K ⊂ R, an estimate M(K)M(K) ≤ c log n (see the definitions in Section 2 below) is an important technical tool intimately related to the Kconvexity constant. It follows by combining works by Lewis and by Figiel and Tomczak-Jaegermann, with deep results of Pisier on Rademacher projections (see e.g., [26]). Although the symmetry can be easily removed from the first two parts, Pisier’s argument use it in a very essential way. In Section 4 we show, in particular, that every convex body K has a position K1 (i.e., K1 = uK − a for some operator u and a ∈ R) such that a random projection, PK1, of dimension [n/2] satisfies M(PK1)M(K 1 ) ≤ C log n, where C is an absolute constant. Moreover, there exists a unitary operator u such that M(K1 +uK1)M(K 1 ) ≤ C log n. Our proof is based essentially on symmetric considerations, a non-symmetric part is reduced to classical facts and simple

Regular ellipsoids and a Blaschke-Santaló-type inequality for projections of non-symmetric convex bodies

Isomorphic properties of intersection bodies

The vertex index of a symmetric convex body \(\mathbf{K} \subset {\mathbb{R}}^{n}\), vein(K), was introduced in [Bezdek, Litvak, Adv. Math. 215, 626–641 (2007)]. Bounds on the vertex index were given in the general case as well as for some basic examples. In this note we improve these bounds and discuss their sharpness. We show that $$\mathrm{vein}(\mathbf{K}) \leq 24{n}^{3/2},$$ which is asymptotically sharp. We also show that the estimate $$\frac{{n}^{3/2}} {\sqrt{2\pi e}\ \mbox{ ovr}(\mathbf{K})} \leq \mathrm{vein}(\mathbf{K}),$$ obtained in [Bezdek, Litvak, Adv. Math. 215, 626–641 (2007)] (here ovr(K) denotes the outer volume ratio of K), is not always sharp. Namely, we construct an example showing that there exists a symmetric convex body K which simultaneously has large outer volume ratio and large vertex index. Finally, we improve the constant in the latter bound for the case of the Euclidean ball from \(\sqrt{2\pi e}\) to \(\sqrt{3}\), providing a completely new approach to the problem.

The simplex was conjectured to be the extremal convex body for the two following problems of asymmetry: (P1) What is the minimal possible value of the quantity $$\max _{K'} |K'|/|K|$$maxK?|K?|/|K|? Here, $$K'$$K? ranges over all symmetric convex bodies contained in K. (P2) What is the maximal possible volume of the Blaschke body of a convex body of volume 1? Our main result states that (P1) and (P2) admit precisely the same solutions. This complements a result from Boroczky et al. (Discrete Math 69:101---120, 1986), stating that if the simplex solves (P1), then the simplex solves (P2) as well.

In 1959 C. A. Rogers gave the following estimate for the density ϑ(k) of lattice-coverings of euclidean d-space Ed with convex bodies . Here, c is a suitable constant which does not depend on d and K. Moreover, Rogers proved that for the unit ball Bd the upper bound can be replaced by , which is, of course a major improvement. In the present paper we show that such an improvement can be obtained for a larger class of convex bodies. In particular, we prove the following theorem. Let K be a convex body in Ed, and let k be an integer satisfying k > log2 loge d + 4. If there exist at least k hyperplanes H1,…, Hk with normals mutually perpendicular and an affine transformation A such that A(K) is symmetrical with respect to Hl,…,Hk, respectively, then . Actually, for a bound of this type we do not even need any symmetry assumption. In fact, some weaker properties concerning shadow boundaries will suffice.

Estimates for measures of lower dimensional sections of convex bodies

A counterexample to a strong variant of the Polynomial Freiman-Ruzsa conjecture in Euclidean space

We propose strongly consistent algorithms for reconstructing the characteristic function $1_K$ of an unknown convex body $K$ in $\mathbb {R}^n$ from possibly noisy measurements of the modulus of its Fourier transform $\widehat {1_K}$. This represents a complete theoretical solution to the Phase Retrieval Problem for characteristic functions of convex bodies. The approach is via the closely related problem of reconstructing $K$ from noisy measurements of its covariogram, the function giving the volume of the intersection of $K$ with its translates. In the many known situations in which the covariogram determines a convex body, up to reflection in the origin and when the position of the body is fixed, our algorithms use $O(k^n)$ noisy covariogram measurements to construct a convex polytope $P_k$ that approximates $K$ or its reflection $-K$ in the origin. (By recent uniqueness results, this applies to all planar convex bodies, all three-dimensional convex polytopes, and all symmetric and most (in the sense of Baire category) arbitrary convex bodies in all dimensions.) Two methods are provided, and both are shown to be strongly consistent, in the sense that, almost surely, the minimum of the Hausdorff distance between $P_k$ and $\pm K$ tends to zero as $k$ tends to infinity.