Simplexes with Prescribed Edge Lengths in Minkowski and Banach Spaces

Abstract

Existence of even a single regular d-simplex in an arbitrary Minkowski space M ^d of dimension d ≧ 4 is questionable. At least, for any d ≧ 4 there is an example of M ^d with four equidistant points in it which cannot be augmented by any fifth such point. At the same time, regular tetrahedra in M ^d, d ≧ 3, and regular triangles in M ^d, d ≧ 2, can be constructed as freely as in E ^d. Suppose that the Banach-Mazur distance δ between the unit balls of M ^d and E ^d satisfies

$$\delta = \delta \left( {M^d } \right) \leqq \delta _1 \left( d \right)\mathop = \limits^{{\text{def}}} \left\{ \begin{gathered} \sqrt {\frac{{d^2 + 2d}}{{d^2 - 2}}} {\text{ for even }}d \geqq 4; \hfill \\ \sqrt {\frac{{d + 1}}{{d - 1}}} {\text{ for odd }}d \geqq 5. \hfill \\ \end{gathered} \right.$$

We prove then that regular d-simplexes in M ^d can be constructed as freely as in E ^d. In fact, a more general theorem dealing with simplexes sufficiently close to regular ones has been proved.

This result can be applied to finite-dimensional subspaces of an infinite-dimensional Banach space X. It is known that, for any d ≧ 2 and any ε > 0, the space X has a d-dimensional subspace M ^d with δ(M ^d) ≦ 1 + ε. Under a proper selection of ε, the condition δ(M ^d) ≦ δ₁(d) above holds which guarantees the existence of regular d-simplexes in M ^d ⊂ X.