Related papers: The generalized Busemann-Petty problem with weight…

The isoperimetric problem with a density or weighting seeks to enclose prescribed weighted area with minimum weighted perimeter. According to Chambers' recent proof of the Log Convex Density Conjecture, for many densities on $\mathbb{R}^n$…

Motivated by the central limit problem for convex bodies, we study normal approximation of linear functionals of high-dimensional random vectors with various types of symmetries. In particular, we obtain results for distributions which are…

We investigate the lower asymptotic density of sumsets in $\mathbb{N}^2$ by proving certain Pl\"unnecke type inequalities for various notions of lower density in $\mathbb{N}^2$. More specifically, we introduce a notion of lower tableaux…

A $\sqrt{n}$ estimate in the hyperplane problem with arbitrary measures has recently been proved in \cite{K3}. In this note we present analogs of this result for sections of lower dimensions and in the complex case. We deduce these…

I discuss gauge and global symmetries in particle physics, condensed matter physics, and quantum gravity. In a modern understanding, global symmetries are approximate and gauge symmetries may be emergent. (Based on a lecture at the April,…

We introduce a new volume definition on normed vector spaces. We show that the induced $k$-area functionals are convex for all $k$. In the particular case $k=2$, our theorem implies that Busemann's 2-volume density is convex, which was…

We propose an approach to infer large-scale heterogeneities within a small celestial body from measurements of its gravitational potential, provided for instance by spacecraft radio-tracking. The non-uniqueness of the gravity inversion is…

We consider the motion of small bodies in general relativity. The key result captures a sense in which such bodies follow timelike geodesics (or, in the case of charged bodies, Lorentz-force curves). This result clarifies the relationship…

Inspired by possible connections between gravity and foundational question in quantum theory, we consider an approach for the adaptation of objective collapse models to a general relativistic context. We apply these ideas to a list of open…

The paper studies possible functional analogs of classical problems from convex geometry. In particular, we provide some bounds in the functional Shephard, Busemann-Petty, and Milman problems generalizing known bounds in this problems for…

We study the Discrete Gauss Image Problem, a generalization of Aleksandrov's classical question on the existence of convex bodies with prescribed integral curvature. We introduce a combinatorial problem called the Assignment Problem and…

For the class of systems of PDEs, for which infinitesimal translations (with respect to some (in)dependent variables) possess specific finite-dimensional invariant subspaces of the space of generalized symmetries of the system considered.…

In this note we consider two topics involving the relationship between the symplectic capacity and the mean width of convex bodies in $\mathbb{R}^{2n}$. We first describe an alternative path from the symplectic Brunn-Minkowski inequality of…

In this paper we prove a series of Rogers-Shephard type inequalities for convex bodies when dealing with measures on the Euclidean space with either radially decreasing densities, or quasi-concave densities attaining their maximum at the…

Magnitude is a numerical invariant of compact metric spaces, originally inspired by category theory and now known to be related to myriad other geometric quantities. Generalizing earlier results in $\ell_1^n$ and Euclidean space, we prove…

We show that the cone-volume measure of a convex body with centroid at the origin satisfies the subspace concentration condition. This implies, among others, a conjectured best possible inequality for the $\mathrm{U}$-functional of a convex…

This paper studies the general Lp dual curvature density equation under a group symmetry assumption. This geometric partial differential equation arises from the general Lp dual Minkowski problem of prescribing the Lp dual curvature measure…

The Loomis-Whitney inequality states that the volume of a convex body is bounded by the product of volumes of its projections onto orthogonal hyperplanes. We provide an extension of both this fact and a generalization of this fact due to…

We study sequential probability assignment in the Gaussian setting, where the goal is to predict, or equivalently compress, a sequence of real-valued observations almost as well as the best Gaussian distribution with mean constrained to a…

Models with extra dimensions have changed our understanding of the hierarchy problem. In general, these models explain the weakness of gravity by diluting gravity in a large bulk volume, or by localizing the graviton away from the standard…