Related papers: On a generalization of Busemann's intersection ine…
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…
Uniformization theory of Gromov hypebolic spaces investigated by Bonk, Heinonen and Koskela, generalizes the case where a classical Poincar\'e ball type model is used as the starting point. In this paper, we develop this approach in the…
We study the volume of the intersection of two unit balls from one of the classical matrix ensembles GOE, GUE and GSE, as the dimension tends to infinity. This can be regarded as a matrix analogue of a result of Schechtman and…
In this paper, the isodiametric problem for centrally symmetric convex bodies in the Euclidean d-space R^d containing no interior non-zero point of a lattice L is studied. It is shown that the intersection of a suitable ball with the…
We give some natural sufficient conditions for balls in a metric space to have small intersection. Roughly speaking, this happens when the metric space is (i) expanding and (ii) well-spread, and (iii) a certain random variable on the…
In the present work we address the issue concerning the relation mass versus radius for bosonic stars, all this in the context of the theory of ultra-cold gases in the dilute regime and within the realm of Newtonian gravity. It will be…
For a symmetric convex body $K\subset\mathbb{R}^n$, the Dvoretzky dimension $k(K)$ is the largest dimension for which a random central section of $K$ is almost spherical. A Dvoretzky-type theorem proved by V.~D.~Milman in 1971 provides a…
A universal inequality that bounds the angular momentum of a body by the square of its size is presented and heuristic physical arguments are given to support it. We prove a version of this inequality, as consequence of Einstein equations,…
Let $\Omega \subset \mathbb{R}^2$ be a convex set. We study the problem of distributing a one-dimensional set $S$ with total length $L$ so that for any line $\ell$ in $\mathbb{R}^2$ the number of intersections $\#(\ell \cap S)$ is…
The Buchdahl limit for static spherically symmetric isotropic stars is generalised to the case of five dimensional Gauss-Bonnet gravity. Our result depends on the sign of the Gauss-Bonnet coupling constant $\alpha$. When $\alpha>0$, we…
In [SZ], Soprunov and Zvavitch have translated the Bezout inequalities (from Algebraic Geometry) into inequalities of mixed volumes satisfied by the simplex. They conjecture this set of inequalities characterizes the simplex, among all…
The theory of dual mixed volumes is extended to star bodies in cotangent bundles and is used to prove several isosystolic inequalities for Hamiltonian systems and Finsler metrics.
In this work we study upper bounds for the ratio of successive inner and outer radii of a convex body K. This problem was studied by Perel'man and Pukhov and it is a natural generalization of the classical results of Jung and Steinhagen. We…
We provide a natural generalization of a geometric conjecture of F\'{a}ry and R\'{e}dei regarding the volume of the convex hull of $K \subset {\mathbb R}^n$, and its negative image $-K$. We show that it implies Godbersen's conjecture…
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 explore for compact Riemannian surfaces whose boundary consists of a single closed geodesic the relationship between orthospectrum and boundary length. More precisely, we establish a uniform lower bound on the boundary length in terms of…
We establish upper bounds for the distance to finite-dimensional subspaces in inner product spaces and improve some generalisations of Bessel's inequality obtained by Boas, Bellman and Bombieri. Refinements of the Hadamard inequality for…
We prove an inequality that extends to arbitrary measures the hyperplane inequality for volume of unconditional convex bodies originally observed by Bourgain.
The classical Busemann-Petty problem asks whether smaller central hyperplane sections of origin-symmetric convex bodies necessarily imply smaller total volume. Zvavitch studied this question for arbitrary measures with continuous even…
We study the lower bound for Koldobsky's slicing inequality. We show that there exists a measure $\mu$ and a symmetric convex body $K \subseteq \mathbb{R}^n$, such that for all $\xi\in S^{n-1}$ and all $t\in \mathbb{R},$…