Related papers: On the Busemann-Petty problem concerning central s…
The Busemann-Petty problem asks whether origin-symmetric convex bodies in $\mathbb{R}^n$ with smaller central hyperplane sections necessarily have smaller $n$-dimensional volume. It is known that the answer is affirmative if $n\le 4$ and…
Since the answer to the complex Busemann-Petty problem is negative in most dimensions, it is natural to ask what conditions on the (n-1)-dimensional volumes of the central sections of complex convex bodies with complex hyperplanes allow to…
The complex Busemann-Petty problem asks whether origin symmetric convex bodies in $\C^n$ with smaller central hyperplane sections necessarily have smaller volume. We prove that the answer is affirmative if $n\le 3$ and negative if $n\ge 4.$
We formulate an isomorphic version of the Busemann-Petty problem and solve it in affirmative in the case of sections of proportional dimensions.
The aim of this paper is to study properties of sections of convex bodies with respect to different types of measures. We present a formula connecting the Minkowski functional of a convex symmetric body K with the measure of its sections.…
The Busemann-Petty problem asks whether origin-symmetric convex bodies in $\mathbb{R}^n$ with smaller central hyperplane sections necessarily have smaller $n$-dimensional volume. It is known that the answer to this problem is affirmative if…
The generalized Busemann-Petty problem asks whether centrally-symmetric convex bodies having larger volume of all m-dimensional sections necessarily have larger volume. When m>3 this is known to be false, but the cases m=2,3 are still open.…
The lower dimensional Busemann-Petty problem asks, whether n-dimensional centrally symmetric convex bodies with smaller i-dimensional central sections necessarily have smaller volumes. The paper contains a complete solution to the problem…
We prove stability in the affirmative part of the Busemann-Petty problem on sections of complex convex bodies.
The Busemann-Petty problem asks whether origin-symmetric convex bodies in real Euclidean n-space with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative for n less or equal to 4 and negative if n…
The generalized Busemann-Petty problem asks whether origin-symmetric convex bodies with lower-dimensional smaller sections necessarily have smaller volume. We study the weighted version of this problem corresponding to the physical…
The Busemann-Petty problem asks whether origin symmetric convex bodies in $\R^n$ with smaller hyperplane sections necessarily have smaller volume. The answer is affirmative if $n\leq 3$ and negative if $n\geq 4.$ We consider a class of…
The lower dimensional Busemann-Petty problem asks whether origin symmetric convex bodies in $\mathbb{R}^n$ with smaller volume of all $k$-dimensional sections necessarily have smaller volume. As proved by Bourgain and Zhang, the answer to…
The complex Busemann-Petty problem asks whether origin symmetric convex bodies in C^n with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative if n\leq 3 and negative if n\geq 4. In this article we…
The classical Busemann-Petty problem (1956) asks, whether origin-symmetric convex bodies in $\mathbb {R}^n$ with smaller hyperplane central sections necessarily have smaller volumes. It is known, that the answer is affirmative if $n\le 4$…
The lower dimensional Busemann-Petty problem asks whether origin-symmetric convex bodies in R^n with smaller volume of all k-dimensional sections necessarily have smaller volume. The answer is negative for k>3. The problem is still open for…
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…
We derive a formula connecting the derivatives of parallel section functions of an origin-symmetric star body in R^n with the Fourier transform of powers of the radial function of the body. A parallel section function (or (n-1)-dimensional…
The Busemann-Petty problem asks whether symmetric convex bodies in the Euclidean space $\mathbb{R}^n$ with smaller central hyperplane sections necessarily have smaller volume. The solution has been completed and the answer is affirmative if…
The aim of this note is to survey the results in some geometric problems related to the centroids and the static equilibrium points of convex bodies. In particular, we collect results related to Gr\"unbaum's inequality and the…