EQUILATERAL DIMENSION OF CERTAIN CLASSES OF NORMED SPACES

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 l(infinity)(n). For smooth and symmetric spaces, Musielak-Orlicz spaces satisfying an additional condition and every (n - 1)-dimensional subspace of l(infinity)(n) 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 l(p)(n) spaces.