Related papers: On the Homothety Conjecture

Let ${\cal K}^n$ be the set of all convex bodies in $\mathbb R^n$ endowed with the Hausdorff distance. We prove that if $K\in {\cal K}^n$ has positive generalized Gauss curvature at some point of its boundary, then $K$ is not a local…

In this erratum, we recover the results from an earlier paper of the author's which contained a gap. Specifically, we prove that if X is a Banach space with an unconditional basis and admits a C^{p}-smooth, Lipschitz bump function, and Y is…

In this note, we show that the strong Viterbo conjecture holds true on any convex toric domain, and that the Viterbo's volume-capacity conjecture holds for the product of a $1$-unconditional convex body $A\subset\mathbb{R}^{n}$ and its…

Let $K$ be a convex body and $K^\circ$ its polar body. Call $\phi(K)=\frac{1}{|K||K^\circ|}\int_K\int_{K^\circ}< x,y>^2 dxdy$. It is conjectured that $\phi(K)$ is maximum when $K$ is the euclidean ball. In particular this statement implies…

The covariogram g_K(x) of a convex body K \subseteq E^d is the function which associates to each x \in E^d the volume of the intersection of K with K+x. Matheron asked whether g_K determines K, up to translations and reflections in a point.…

Let $n\geq C$ for a large universal constant $C>0$, and let $B$ be a convex body in $R^n$ such that for any $(x_1,x_2,\dots,x_n)\in B$, any choice of signs $\varepsilon_1,\varepsilon_2,\dots,\varepsilon_n\in\{-1,1\}$ and for any permutation…

In this work we discuss a conjecture of Viterbo relating the symplectic capacity of a convex body and its volume. The conjecture states that among all 2n-dimensional convex bodies with a given volume the euclidean ball has maximal…

Let $K$ be a convex body in $\mathbb{R}^n$ and $f : \partial K \rightarrow \mathbb{R}_+$ a continuous, strictly positive function with $\int\limits_{\partial K} f(x) d \mu_{\partial K}(x) = 1$. We give an upper bound for the approximation…

High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to the sphere and the hyperbolic…

We show that the rate of convergence on the approximation of volumes of a convex symmetric polytope P in R^n by its dual L_{p$-centroid bodies is independent of the geometry of P. In particular we show that if P has volume 1,…

The Brunn-Minkowski Theorem asserts that $\mu_d(A+B)^{1/d}\geq \mu_d(A)^{1/d}+\mu_d(B)^{1/d}$ for convex bodies $A,\,B\subseteq \R^d$, where $\mu_d$ denotes the $d$-dimensional Lebesgue measure. It is well-known that equality holds if and…

Let $E_1,E_2\subset \mathbb{R}^n$ be two homothetic solid ellipsoids, $n\geq 3$, with center at the origin $O$ of a system coordinates of $\mathbb{R}^n$, and $E_1\subset E_2$. Then there exists a $O$-symmetric ellipsoid $E_3$ such that…

Let $\mu$ be a probability measure on a separable Banach space $X$. A subset $U\subset X$ is $\mu$-continuous if $\mu(\partial U)=0$. In the paper the $\mu$-continuity and uniform $\mu$-continuity of convex bodies in $X$, especially of…

If $K\subset\mathbb{R}^n$ is a convex body and $\Gamma_pK$ is the $p$-centroid body of $K$, the $L_p$ Busemann-Petty centroid inequality states that $\vol(\Gamma_pK) \geq \vol(K)$, with equality if and only if $K$ is an ellipsoid centered…

Let us consider the set of all joint probabilities generated by local binary measurements on two separated quantum systems of a given local dimension d. We address the question of whether the shape of this quantum body is convex or not. We…

It is known that for a convex body K in R^d of volume one, the expected volume of random simplices in K is minimised if K is an ellipsoid, and for d = 2, maximised if K is a triangle. Here we provide corresponding stability estimates.

Suppose that $K \subseteq \RR^d$ is a 0-symmetric convex body which defines the usual norm $$ \Norm{x}_K = \sup\Set{t\ge 0: x \notin tK} $$ on $\RR^d$. Let also $A\subseteq\RR^d$ be a measurable set of positive upper density $\rho$. We show…

We formulate a generalization of the volume conjecture for planar graphs. Denoting by <G, c> the Kauffman bracket of the graph G whose edges are decorated by real "colors" c, the conjecture states that, under suitable conditions, certain…

A 1972 duality conjecture due to Pietsch asserts that the entropy numbers of a compact operator acting between two Banach spaces and those of its adjoint are (in an appropriate sense) equivalent. This is equivalent to a dimension free…

Let $K$ be a smooth convex set with volume one in $\BBR^d$. Choose $n$ random points in $K$ independently according to the uniform distribution. The convex hull of these points, denoted by $K_n$, is called a {\it random polytope}. We prove…