度量几何

We introduce the Point-Cross Dimension, a new pointwise invariant designed to measure the directional organization of a set at a single point. Whereas the Point-Extended Box Dimension quantifies local dispersion and covering complexity, the…

We prove that the only planar, centrally symmetric, strictly convex body $K\subset\mathbb{R}^2$ with $C^1$ boundary that floats in equilibrium in every orientation for the perimetral densities $\sigma=\tfrac18$ or $\sigma=\tfrac38$ is a…

In this paper, a residual-safeguarded local Matrix Fisher--Gaussian (MFG) inference method is developed for wrench-based pose estimation on $\mathrm{SO}(3)\times\mathbb{R}^3$. The force/torque measurements are modeled by a quasi-static…

We introduce the microscopic weighting, a canonical signed measure of mass one that can be associated to almost any finite metric space. The microscopic weighting is obtained as the small-scale limit of the weightings used to define the…

We study the $L^p(\mu)$-norm of the metric projection onto a closed, convex set $C \subset \mathbf{R}^n$ when $\mu$ is the uniform measure on the sphere or the standard Gaussian measure on $\mathbf{R}^n$. Up to universal constants, we…

Let $\lambda_j(K)$ be the $j$th Dirichlet eigenvalue of a convex body $K$. It is well known that $\lambda_1$ satisfies a Brunn--Minkowski inequality: $K \mapsto \lambda_1(K)^{-1/2}$ is concave on the family of convex bodies. We show that no…

For compact convex sets $L,K \subset \mathbb{R}^n$, denote by $\lambda_K(L)$ the smallest size of a homothet of $K$ that contains $L$. We define a measure of symmetry based on the $n$-simplex $\Delta = \Delta^n \subset \mathbb{R}^n$ as the…

We study generalizations of the classical Rogers--Shephard inequalities in the framework of Firey $L_p$-summation. We first consider the class of asymmetric $L_p$-zonoids. In this setting, we show that proving a sharp $L_p$-Rogers--Shephard…

We give a novel proof of the fact that every coarsely separating family of subsets of the Euclidean space $\mathbb{R}^{d}$ must have asymptotic dimension at least $d-1$. The proof only uses singular homology/cohomology and standard facts…

We study preimage regions on the open probability simplex associated with symmetric separable functionally generated maps. The problem is a finite-dimensional geometric question about convexity and barycentric star-shapedness of these…

This paper establishes a unified Minkowski theory for exterior p-capacitary volumes and resolves the classical P\'olya-Szeg\"o conjecture on the electrostatic capacity of convex bodies.

A crystallographic group is a discrete subgroup of the Euclidean group $\operatorname{E}(n)$ that has a compact fundamental domain. Since such a crystallographic group $\Gamma$ is infinite, computing fundamental domains of $\Gamma$ is…

A long-standing conjecture of Lapidus asserts that under certain conditions a self-similar fractal set is not Minkowski measurable if and only if it is of lattice-type. For self-similar sets in $\mathbb{R}$, the Lapidus conjecture has been…

We investigate the $L_p$ Brunn-Minkowski inequality for dual quermassintegrals in weighted measure spaces, which is a special class of rotationally invariant measures proposed by Cordero-Erausquin and Rotem [Ann. Probab., {\bf 51} (2023)].…

The Blaschke-Lebesgue theorem states that the Reuleaux triangle has the smallest area among planar convex bodies of a fixed constant width. We study how small bodies of constant width can be on the unit sphere $\mathbb S^n$ when $n$ is…

In this paper, we systematically investigate the structural and operator-theoretic properties of mappings contracting perimeters of triangles (MCPTs) within the generalized topological framework of complete $b$-metric spaces with…

We introduce a discrete-energy Sobolev space $\mathcal{W}^{1,p}_{\mathscr V}(T)$ on Ahlfors regular snowtrees, a class of metric trees where every arc is a snowflake of the same type. Our main result shows that, for every partition…

This is a geometric retelling of Konyagin and Sevast'yanov's proof of Andrew's theorem, which is a tight upper bound on the number of vertices of a d-dimensional lattice polytope in terms of its volume.

We introduce and study peeling and wrapping operations for families of compact convex sets. The two peeling procedures considered in the paper are the $m$-point peeling, obtained by intersecting the convex hulls remaining after all possible…

A complete classification of weak$^*$~continuous, measure-valued valuations is established on star bodies in $\R^n$. Consequences are an integral representation of rotation equivariant, measure-valued valuations and a characterization of…