Related papers: The modified complex Busemann-Petty problem on sec…

H. Busemann and C. M. Petty posed the following problem in 1956: If K and L are origin-symmetric convex bodies in R^n and for each hyperplane H through the origin the volumes of their central slices satisfy vol(K cap H) < vol(L cap H), does…

We provide an affirmative answer to a variant of the Busemann-Petty problem, proposed by V.~Milman: Let $K$ be a convex body in ${\mathbb R}^n$ and let $D$ be a compact subset of ${\mathbb R}^n$ such that, for some $1\ls k\ls n-1$,…

We introduce complex intersection bodies and show that their properties and applications are similar to those of their real counterparts. In particular, we generalize Busemann's theorem to the complex case by proving that complex…

This article belongs to the area of geometric tomography, which is the study of geometric properties of solids based on data about their sections and projections. We describe a new direction in geometric tomography where different…

We study the slicing inequality for the surface area instead of volume. This is the question whether there exists a constant $\alpha_n$ depending (or not) on the dimension $n$ so that $$S(K)\leq\alpha_n|K|^{\frac{1}{n}}\max_{\xi\in…

We prove that the (n-2)-dimensional surface area (perimeter) of central hyperplane sections of the n-dimensional unit cube is maximal for the hyperplane perpendicular to the vector (1,1,0,...,0). This gives a positive answer to a question…

We prove new versions of the isomorphic Busemann-Petty problem for two different measures and show how these results can be used to recover slicing and distance inequalities. We also prove a sharp upper estimate for the outer volume ratio…

We consider a generalization of the hyperplane problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies and for duals of bodies with bounded volume…

Several years ago the authors started looking at some problems of convex geometry from a more general point of view, replacing volume by an arbitrary measure. This approach led to new general properties of the Radon transform on convex…

In this paper we study how certain symmetries of convex bodies affect their geometric properties. In particular, we consider the impact of symmetries generated by the block diagonal subgroup of orthogonal transformations, generalizing…

We study a version of the Busemann-Petty problem for $\log$-concave measures with an additional assumption on the dilates of convex, symmetric bodies. One of our main tools is an analog of the classical large deviation principle applied to…

We prove the following theorem. Let $\mu$ be a measure on $R^n$ with even continuous density, and let $K,L$ be origin-symmetric convex bodies in $R^n$ so that $\mu(K\cap H)\le \mu(L\cap H)$ for any central hyperplane H. Then $\mu(K)\le…

We show that in all dimensions d>2, there exists an asymmetric convex body of revolution all of whose maximal hyperplane sections have the same volume. This gives the negative answer to the question posed by V. Klee in 1969.

We prove several estimates for the moments of arbitrary measures on convex bodies. We apply these estimates to show a new slicing inequality for measures on convex bodies. We also deduce estimates for the outer volume ratio distance from an…

The main goal of this paper is to present a series of inequalities connecting the surface area measure of a convex body and surface area measure of its projections and sections. We present a solution of a question from S. Campi, P.…

This note is motivated by the article of Bamerni, Kadets and Kili\c{c}man [J. Math. Anal. Appl. 435 (2), 1812--1815 (2016)]. We consider the remaining problem which claims that if $A$ is a dense subset of a finite dimensional space $X$,…

Let $K$ be a convex body in $\mathbb R^n$. We prove that in small codimensions, the sections of a convex body through the centroid are quite symmetric with respect to volume. As a consequence of our estimates we give a positive answer to a…

We classify the full set of convex central configurations in the Newtonian four-body problem. Particular attention is given to configurations possessing some type of symmetry or defining geometric property. Special cases considered include…

We give obstructions for a noncompact manifold to admit a complete Riemannian metric with (nonuniformly) positive scalar curvature. We treat both the finite volume and infinite volume cases.

The convex shape contained in a disk having prescribed area and maximal perimeter is completely characterized in terms of the area fraction. The solution is always a polygon having all but one sides equal. The lengths of the sides are…