A set in a normed space is equilateral if any two points of are at the same distance:
There exists such that for all distinct .
From now on we fix by scaling.
If is -dimensional, then an equilateral set has at most elements (Petty 1971, Soltan 1975). The following is a very simple proof. Let be the convex hull of . Then the homothets , are all contained in , and have pairwise disjoint interiors. This follows from the triangle inequality. Now use the fact that the Lebesgue measure of each homothet is times that of .
The next proof is by Füredi, Lagarias and Morgan (1993). Let , the union of all balls of radius centered at the points in . Then has diameter 2. Now, by the isodiametric inequality (Busemann 1947, Mel’nikov 1963), the unit ball is the set of diameter 2 of largest Lebesgue volume. Again comparing volumes gives the upper bound.
The isodiametric inequality has a simple proof from the Brunn-Minkowski inequality, so it is not surprising that there is also a proof using the latter inequality. Again consider the set . Since it has diameter 2, its central symmetral is contained in a ball of radius 2. By the Brunn-Minkowski inequality, , and the upper bound follows.
- H. Busemann, Intrinsic area, Ann. Math. 48 (1947), 234-267. [It is quite a challenge to find the isodiametric inequality in this paper. It’s equation (2.2) on p. 241.]
- Z. Füredi, J. C. Lagarias, and F. Morgan, Singularities of minimal surfaces and networks and related extremal problems in Minkowski space, Discrete and computational geometry (New Brunswick, NJ, 1989/1990), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 6, Amer. Math. Soc., Providence, RI, 1991. pp. 95-109.
- M. S. Mel’nikov, Dependence of volume and diameter of sets in an n-dimensional Banach space (Russian), Uspehi Mat. Nauk 18 (1963), 165-170.
- C. M. Petty, Equilateral sets in Minkowski spaces, Proc. Amer. Math. Soc. 29 (1971), 369-374.
- P. S. Soltan, Analogues of regular simplexes in normed spaces (Russian), Dokl. Akad. Nauk SSSR 222 (1975), 1303-1305. English translation: Soviet Math. Dokl. 16 (1975), 787-789.