On the multiple Borsuk numbers of sets

TL;DR

This paper introduces the concept of k-fold Borsuk numbers, generalizing the classic Borsuk number, and provides characterizations and bounds for various types of sets in Euclidean spaces.

Contribution

It defines and analyzes the k-fold Borsuk number, extending existing theory to new classes of sets and offering bounds and characterizations in Euclidean geometry.

Findings

Characterization of k-fold Borsuk numbers in the Euclidean plane

Bounds for centrally symmetric, smooth, and convex bodies

Analysis of k-fold Borsuk numbers for finite point sets in 3D

Abstract

The Borsuk number of a set S of diameter d >0 in Euclidean n-space is the smallest value of m such that S can be partitioned into m sets of diameters less than d. Our aim is to generalize this notion in the following way: The k-fold Borsuk number of such a set S is the smallest value of m such that there is a k-fold cover of S with m sets of diameters less than d. In this paper we characterize the k-fold Borsuk numbers of sets in the Euclidean plane, give bounds for those of centrally symmetric sets, smooth bodies and convex bodies of constant width, and examine them for finite point sets in the Euclidean 3-space.