The precised equilateral problem in Minkovsky space
by
Vladimir V. Makeev
St. Petersburg State University

It is unknown for n \geqslant 4 whether each n-dimensional normed space contains n+1 points at pairwise distance one.

Let's consider the following precised problem. Whenever for each two norms ||      ||₁ and ||      ||₂ in Rⁿ this space contains such n+1 points A₁, ..., A_n+1, that all parwise distances || A_iA_j||₁=1, 1\leqslant i < j\leqslant n+1 and parwise distances || A_iA_j||₂ reach not more then n values. The following theorem gives an affirmative answer on this question for n=3.

Date received: August 11, 1999