度量几何

We prove that convexity of the Boltzmann entropy at Wasserstein barycenters is strong enough to distinguish Hilbert spaces from general Banach spaces. Thus Wasserstein barycenters provide an intrinsic optimal-transport test for Hilbertian…

We prove that for any matrix $A \in \mathbb{R}^{m \times n}$ and any $\varepsilon \in (0, 1/2]$ there is a diagonal matrix $D \in \mathbb{R}_{\geq 0}^{m \times m}$ with at most $O(\frac{n}{\varepsilon^2} \log(\frac{1}{\varepsilon}))$…

We prove that, on a regular local $p$-Dirichlet space supporting a $p$-Poincar\'e inequality, if the $p$-walk dimension is strictly greater than $p$, then every curve family has zero $p$-modulus. As a consequence, we show that no…

For a Hausdorff topology on the set of ideals of the semilattice $M(X)$ of coarse equivalence classes of metrics on a set $X$, the space $I(M(X))$ of ideals is the closure of the set of principal ideals, thus allowing to view non-principal…

In this paper we deal with Nil geometry, in whose projective model the geodesic curves are helix-like and fit onto geodesic Nil cylinders of revolution with fibrum axes. In this paper we investigate relations for geodesic cylinders and thus…

This paper investigates the topological properties of intersections of balls in finite-dimensional normed spaces - a problem that naturally arises when constructing covers for estimating the Gromov-Hausdorff distance. We study the topology…

We generalize the classical Ceva's and Menelaus's theorems to curvilinear triangles bounded by circular arcs. We introduce trilinear coordinates associated with such triangles and develop several geometric constructions. In particular, for…

In this paper we prove the WALA conjecture.

In this paper, we study the dual Minkowski problem under group symmetry. For $0<q\le n$, we give a complete existence characterization in the framework of $G$-invariant convex bodies when the group $G\subset O(n)$ has no nonzero fixed…

This paper gives a complete description of the solutions of the global positioning problem, emphasizing the under-determined case. We show that the solutions form a quadric, which may degenerate in various ways. Perhaps more surprisingly,…

We define a distance analogous to the Gromov-Hausdorff distance that enables the comparison of arbitrary quasi-isometric spaces. We also investigate properties preserved under limits with respect to this distance, as well as properties of…

We study the bi-Lipschitz embedding problem for a class of metric spaces called slit carpets. First we show that the $n$th stage $\mathbb{M}_n$ of the standard slit carpet of Merenkov admits a bi-Lipschitz embedding into Euclidean space…

The chord-power integrals $I_k$ are classical integral-geometric functionals of a planar convex body, obtained by integrating powers of the chord length against the kinematic measure on the space of lines meeting the body. We establish a…

The study of bodies of constant width is a classical subject in convex geometry, with the 3-dimensional Meissner bodies being canonical examples. This paper presents a novel geometric construction of a body of constant width in $\mathbb…

Finite automata were used to determine multiple addresses in number systems and to find topological properties of self-affine tiles and finite type fractals. We join these two lines of research by axiomatically defining automata which…

The sliced Wasserstein distance $SW_2(\mu,\nu)$ compares high-dimensional probability measures by averaging one-dimensional optimal transport distances over linear projections. Although sliced Wasserstein distances are now standard…

We prove quantitative stability estimates for Wasserstein barycenters on Alexandrov spaces with curvature bounded from below. The proof combines the variational strategy of Carlier--Delalande--M\'erigot with heat-kernel regularization,…

Motivated by a discrete inequality problem proposed by Duanyang Zhang as Problem 6 of the 2022 Spring NSMO, we prove a median version of Hardy's inequality. For a nonnegative function $f\in L^p(0,\infty)$, $p>1$, let $A(t)$ be the average…

We give a simple computational approach to mathematical quasicrystals, combining cut-and-project methods with self-similarity. Starting with a Pisot unit $\beta$ and an iterated function system $g_k(z)=\beta z +z_k, \ k=1,...,m$ in a…

We extend the notion of illumination bodies to Riemannian spaces of constant curvature and to projective Finsler geometries. We prove that the derivative of their volume defines a notion of surface area for convex bodies in these settings,…