Segments and Convexity in Metric Spaces

Tian Vlasic

Home/Archive/arXiv

arXiv · 2607.27222

Segments and Convexity in Metric Spaces

Tian Vlasic

View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper aims both to provide a unified introduction to d-convexity and to contribute new structural results on metric segments and convexity in metric spaces. After developing the basic theory of metric segments, we study their geometric and topological properties, establishing several structural results, and show that metric segments realize arbitrary closed subsets after the metric is replaced by a topologically equivalent and bounded one. We then examine metrically, Menger, and strictly convex spaces, obtaining new characterizations of strict convexity in terms of an order structure on metric segments and their isometric embeddability into R. Additionally, we offer an exposition of the relationship between d-convexity and other notions of convexity in metric spaces from the standpoint of axiomatic convexity. Finally, we characterize metric spaces in which metric segments are trivial via a "local snowflaking" condition.

Explore related subjects

math.MG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Tian Vlasic. 2026-06-30. Segments and Convexity in Metric Spaces. https://arxiv.org/abs/2607.27222

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

$β$-Uniform Convexity and Divisible Domains

Divisible convex sets have long been important in the study of Hilbert geometries. When a divisible convex set is an ellipsoid, the Hilbert geometry it induces is the hyperbolic space. In general, strictly convex divisible domains exhibit negative curvature properties, but only the ellipsoid is a CAT(0) space. The notion of p-uniform convexity from the theory of Banach spaces has been proposed by Shin-Ichi Ohta as a generalization of the Alexandrov-Toponogov comparison theorems to Finsler manifolds. We prove that a natural Finsler metric on a strictly convex divisible domain is $β$-uniformly convex, where the constant $β$ is related to the regularity of the boundary. We use this to show, with AI assistance, that the Hilbert metric, under suitable local and scale-dependent assumptions, is $β$-uniformly convex on such domains.

math.MG↗

arXiv2026

A positive solution to the $L^p$ projection centroid conjecture

In a classical paper [21] in 2000, Lutwak-Yang-Zhang established the $L^p$ analog of the Petty projection inequality and the $L^p$ analog of the Busemann-Petty centroid inequality. In Section 7 of [21], Lutwak-Yang-Zhang proposed the important $L^p$ projection centroid conjecture. We give a positive solution to the $L^p$ projection centroid conjecture in this work.

math.MG↗

arXiv2026

Minimal central slices of the regular simplex

We prove that minimal-volume hyperplane sections of the regular simplex through its centroid are parallel to a facet. The proof combines variational methods with Fourier-analytic techniques and zero-diminishing arguments to show that every critical normal vector has at most three distinct non-zero coordinates. Analysis of the two- and three-value cases then yields the sharp lower bound.

math.MG↗