Related papers: On Ball's conjectured Santal\'o type inequality

The Banach isometric conjecture asserts that a normed space with all of its $k$-dimensional subspaces isometric, where $k\geq 2$, is Euclidean. The first case of $k=2$ is classical, established by Auerbach, Mazur and Ulam using an elegant…

We prove a generalization of the hyperplane inequality for intersection bodies, where volume is replaced by an arbitrary measure $\mu$ with even continuous density and sections are of arbitrary dimension $n-k,\ 1\le k <n.$ If $K$ is a…

We explore the notion of Santal\'o point for the Holmes-Thompson boundary area of a convex body in a normed space. In the case where the norm is $C^1$, and in the case where unit ball and convex body coincide, we prove existence and…

We prove that for any two centrally-symmetric convex shapes $K,L \subset \mathbb{R}^2$, the function $t \mapsto |e^t K \cap L|$ is log-concave. This extends a result of Cordero-Erausquin, Fradelizi and Maurey in the two dimensional case.…

Gr\"unbaum's inequality guarantees that the centroid of a convex body has halfspace depth at least $1/e$: every halfspace containing the centroid captures at least a $1/e$ fraction of the body's volume. For mixed-integer convex sets…

Approximating convex bodies is a fundamental question in geometry, which has a wide variety of applications. Given a convex body $K$ in $\textbf{R}^d$ for fixed $d$, the objective is to minimize the number of facets of an approximating…

In this paper, we present sharp stability results for various reverse isoperimetric problems in $\mathbb R^2$. Specifically, we prove the stability of the reverse isoperimetric inequality for $\lambda$-convex bodies -- convex bodies with…

Let $ K $ be a convex body in $ \mathbb{R}^n $. We denote the volume of $ K $ by $ \vert K\vert $, and the polar body of its difference body $ K - K $ by $ (K - K)^{\circ} $. We provide a new proof of the well-known estimate \[ |K||(K -…

In 1921 Blichfeldt gave an upper bound on the number of integral points contained in a convex body in terms of the volume of the body. More precisely, he showed that $#(K\cap\Z^n)\leq n! \vol(K)+n$, whenever $K\subset\R^n$ is a convex body…

We consider the following measure of symmetry of a convex n-dimensional body K: $\rho(K)$ is the smallest constant for which there is a point x in K such that for partitions of K by an n-1-dimensional hyperplane passing through x the ratio…

We explore alternative functional or transport-entropy formulations of the Blaschke-Santal{\'o} inequality and of its conjectured counterpart due to Mahler. In particular, we obtain new direct and reverse Blaschke-Santal{\'o} inequalities…

We determine when a convex body in $\mathbb{R}^d$ is the closed unit ball of a reasonable crossnorm on $\mathbb{R}^{d_1}\otimes\cdots\otimes\mathbb{R}^{d_l},$ $d=d_1\cdots d_l.$ We call these convex bodies "tensorial bodies". We prove that,…

Let $K \subset \mathbb R^n$ be a convex body with barycenter at the origin. We show there is a simplex $S \subset K$ having also barycenter at the origin such that $\left(\frac{vol(S)}{vol(K)}\right)^{1/n} \geq \frac{c}{\sqrt{n}},$ where…

If a convex body $K \subset \mathbb{R}^n$ is covered by the union of convex bodies $C_1, \ldots, C_N$, multiple subadditivity questions can be asked. Two classical results regard the subadditivity of the width (the smallest distance between…

We study the asymptotic behavior of smooth, origin-symmetric, strictly convex bodies under the centro-affine normal flows. By means of a stability version of the Blaschke-Santal\'{o} inequality, we obtain regularity of the solutions…

The illumination conjecture is a classical open problem in convex and discrete geometry, asserting that every compact convex body~$K$ in $\mathbb R^n$ can be illuminated by a set of no more than $2^n$ points. If $K$ has smooth boundary, it…

This paper is dedicated to study the sine version of polar bodies and establish the $L_p$-sine Blaschke-Santal\'{o} inequality for the $L_p$-sine centroid body. The $L_p$-sine centroid body $\Lambda_p K$ for a star body…

We adapt an argument of Tao and Vu to show that if $\lambda_1\le\cdots\le\lambda_d$ are the successive minima of an origin-symmetric convex body $K$ with respect to some lattice $\Lambda<\mathbb{R}^d$, and if we set…

We prove the non-symmetric Mahler conjecture in dimension three. More precisely, we prove the sharp lower bound \[ \mathcal P(K) \geq \frac{64}{9} \] for every convex body $K \subset \mathbb R^3$, where $\mathcal P(K)$ denotes the…

We derive lower estimates for the approximation of the $d$-dimensional Euclidean ball by polytopes with a fixed number of $k$-dimensional faces, $k\in\{0,1,\ldots,d-1\}$. The metrics considered include the intrinsic volume difference and…