Related papers: On a problem of Turan about positive definite func…

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.…

We study the local and global versions of the convexity, which is closely related to the problem of extending a convex function on a non-convex domain to a convex function on the convex hull of the domain and beyond the convex hull. We also…

We prove upper and lower bounds for a variational functional for convex functions satisfying certain boundary conditions on a sector of the unit ball in two dimensions. The functional contains two terms: The full Hessian and its…

An irregular vertex in a tiling by polygons is a vertex of one tile and belongs to the interior of an edge of another tile. In this paper we show that for any integer $k\geq 3$, there exists a normal tiling of the Euclidean plane by convex…

Given an entire transcendental function f with a non-completely invariant Baker domain, we define a Baker lamination on geodesics to study the divergence and convergence of a pinching process of curves in U. If the boundary of some curve in…

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…

We study how the existence of a negatively pinched K\"ahler metric on a domain in complex Euclidean space restricts the geometry of its boundary. In particular, we show that if a convex domain admits a complete K\"ahler metric, with pinched…

We give an optimal bound on normal curvatures of immersed n-torus in a Euclidean ball of large dimension.

One of the most important problems in Geometric Tomography is to establish properties of a given convex body if we know some properties over its sections or its projections. There are many interesting and deep results that provide…

Let $D$ be a bounded domain $D$ in $\mathbb R^n $ with infinitely smooth boundary and $n$ is odd. We prove that if the volume cut off from the domain by a hyperplane is an algebraic function of the hyperplane, free of real singular points,…

In this paper, we investigate an overdetermined boundary value problem of divergence type on bounded domains in Riemannian manifolds with non-negative Ricci curvature. Using integral identities and the $P$-function method, we derive…

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 article, we study domains $\Omega \subset \mathbb{S}^2$ that support positive solutions of the overdetermined problem $$ \Delta u + f(u,|\nabla u|)=0 \quad \text{in } \Omega, $$ subject to the boundary conditions $u=0$ on…

We answer in the negative a question by Gruenbaum who asked if there exists a finite basis of affine invariant points. We give a positive answer to another question by Gruenbaum about the "size" of the set of all affine invariant points.…

Given a convex body K in R^n and p in R, we introduce and study the extremal inner and outer affine surface areas IS_p(K) = sup_{K'\subseteq K} (as_p(K') ) and os_p(K)=inf_{K'\supseteq K} (as_p(K') ), where as_p(K') denotes the L_p-affine…

In 1969, Vic Klee asked whether a convex body is uniquely determined (up to translation and reflection in the origin) by its inner section function, the function giving for each direction the maximal area of sections of the body by…

Let $K$ be an $n$-dimensional symmetric convex body with $n \ge 4$ and let $K\dual$ be its polar body. We present an elementary proof of the fact that $$(\Vol K)(\Vol K\dual)\ge \frac{b_n^2}{(\log_2 n)^n},$$ where $b_n$ is the volume of the…

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…

In this paper we study the class of so called `ball-bodies' in ${\mathbb R}^n$, given by intersections of translates of Euclidean unit balls (or, equivalently, summand of the Euclidean ball). We study the class along with the natural…

We introduce a generalization of the Euclidean algorithm for rings equipped with an involution, and completely enumerate all isomorphism classes of orders over definite, rational quaternion algebras equipped with an orthogonal involution…