Equilateral Dimension of Certain Classes of Normed Spaces

Abstract

The equilateral dimension of a normed space is the maximal number of pairwise equidistant points of this space. The aim of this article is to study the equilateral dimension of certain classes of finite-dimensional normed spaces. A well-known conjecture states that the equilateral dimension of any n-dimensional normed space is not less than n + 1. By using an elementary continuity argument, we establish it in the following classes of spaces: permutation-invariant spaces, Musielak-Orlicz spaces, and one codimensional subspaces of . For smooth and symmetric spaces, Musielak-Orlicz spaces satisfying an additional condition and every (n − 1)-dimensional subspace of we also provide some weaker bounds on the equilateral dimension for every space that is sufficiently close to one of these. This generalizes a result of Swanepoel and Villa concerning the spaces.

Keywords:

• Brouwer fixed point theorem
• Equilateral dimension
• Equidistant points
• Equilateral set
• Orlicz space
• Symmetric norm
• Touching translates

Mathematics Subject Classification:

• Primary 46B85
• Secondary 46B20
• 52C17
• 52A15
• 52A20

ACKNOWLEDGMENTS

The author thanks the anonymous referees for very careful proofreading and suggestions which significantly improved quality of the paper.