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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What do you think of the following approach to proving Kusner’s conjecture that an upper bound for the size of an equilateral set in is ?

Let denote the 2-adic valuation, the corresponding local ring, its unique maximal ideal and the reduction of modulo . One can prove that there are no equilateral sets with respect to the norm with in as follows:

Since , and hence for any . Therefore, if are points in an equilateral subset of , the image modulo of equals the image modulo of , which equals when and are distinct, and otherwise.

Therefore, if are the points in an equilateral set of size in , then the -matrix

has -entry and is hence equal to the matrix with main diagonal entries all equal to and all other entries equal to . Since this matrix has rank and is equal to the sum of the rank--matrices , we have that .

For a proof by contradiction of Kusner’s conjecture, it would therefore suffice to show that the existence of an equilateral set (with respect to the norm) of size in implies the existence of an equilateral set (with respect to the norm) of size in . This sounds like the sort of thing tropical geometry was invented for, but I don’t understand much tropical geometry, so I’m sort of stuck here. I apologize for dumping this text on you, but I’m just a math-obsessed pupil and not (yet) a professional mathematician, so I just don’t know what to do with this observation.