On covering a ball by congruent subsets in normed spaces

TL;DR

This paper investigates how a ball in certain normed spaces can be covered by congruent subsets, revealing conditions on the center's interior membership relative to the sets and providing specific examples.

Contribution

It establishes a relationship between the number of congruent subsets covering a ball and the position of the center in relation to these sets in normed spaces.

Findings

If the number of sets is not greater than the space's dimension, the center either belongs to the interior of all sets or none.

Examples are provided where the center belongs to the interior of exactly one set.

The results are specific cases related to a modified dissection problem.

Abstract

We consider the covering of a ball in certain normed spaces by its congruent subsets and show that if the finite number of sets is not greater than the dimensionality of the space, then the centre of the ball either belongs to the interior of each set, or doesn't belong to the interior of any set. We also provide some examples when it belongs to the interior of exactly one set. These are the specific cases of the modified problem originally posed for dissection.