Geometric coincidence results from multiplicity of continuous maps

TL;DR

This paper investigates geometric coincidence problems related to convex bodies and hyperplane sections, using topological methods to estimate the multiplicity of continuous maps between manifolds and deriving geometric implications.

Contribution

It introduces topological estimates for map multiplicity and explores their geometric consequences in convex geometry problems.

Findings

Topological estimates for map multiplicity are established.

Examples demonstrate geometric consequences of these estimates.

Results connect topological multiplicity with convex geometric problems.

Abstract

In this paper we study geometric coincidence problems in the spirit of the following problems by B. Gr\"unbaum: How many affine diameters of a convex body in $\mathbb R^n$ must have a common point? How many centers (in some sense) of hyperplane sections of a convex body in $\mathbb R^n$ must coincide? One possible approach to such problems is to find topological reasons for multiple coincidences for a continuous map between manifolds of equal dimension. In other words, we need topological estimates for the multiplicity of a map. In this work examples of such estimates and their geometric consequences are presented.