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.

**References**

- 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.

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Somehow i missed the point. Probably lost in translation 🙂 Anyway … nice blog to visit.

cheers, Correspond!

Hi, Correspond,

Thank you. The point is just to review three proofs that there are at most equidistant points in an -dimensional normed space.

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