Home » Discrete Geometry » Equilateral sets in normed spaces: upper bounds

# Equilateral sets in normed spaces: upper bounds

A set $S$ in a normed space $X$ is equilateral if any two points of $S$ are at the same distance:

There exists $\rho>0$ such that $\|x-y\|=\rho$ for all distinct $x,y\in S$.

From now on we fix $\rho=1$ by scaling.

If $X$ is $n$-dimensional, then an equilateral set has at most $2^n$ elements (Petty 1971, Soltan 1975). The following is a very simple proof. Let $P$ be the convex hull of $S$. Then the homothets $x+\frac12(P-x), x\in S$, are all contained in $P$, and have pairwise disjoint interiors. This follows from the triangle inequality. Now use the fact that the Lebesgue measure of each homothet is $(1/2)^n$ times that of $P$.

The next proof is by Füredi, Lagarias and Morgan (1993). Let $A=\bigcup_{x\in S} B(x,\frac12)$, the union of all balls of radius $\frac12$ centered at the points in $S$. Then $A$ 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 $A$. Since it has diameter 2, its central symmetral $A-A:=\{x-y : x,y\in A\}$ is contained in a ball of radius 2. By the Brunn-Minkowski inequality, $\mu(A-A)^{1/n}\geq\mu(A)^{1/n}+\mu(-A)^{1/n}$, and the $2^n$ upper bound follows.

### References

1. 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.]
2. 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.
3. 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.
4. C. M. Petty, Equilateral sets in Minkowski spaces, Proc. Amer. Math. Soc. 29 (1971), 369-374.
5. 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.
Thank you. The point is just to review three proofs that there are at most $2^n$ equidistant points in an $n$-dimensional normed space.