How many points in Euclidean space can have a common nearest neighbor?

An Euclidean code is a finite set of codepoints in n-dimensional Euclidean space, ℛⁿ. The total number of nearest neighbors of a given codepoint in the code is called its touching number. We show that the maximum number of codepoints F_n that can share the same nearest neighbor codepoint is equal to the maximum kissing number τ_n in n dimensions, that is, the maximum number of unit spheres that can touch a given unit sphere without overlapping. We then apply a known upper bound on τ_n to obtain F_n⩽2^{n(0.401+&ogr;(1))}, which improves upon the best known upper bound of F_n⩽2^{n(1+&ogr;(1))} . We also show that the average touching number of all the points in an Euclidean code is upper bounded by τ_n