On the Volume of Finite Arrangements of Spheres

Abstract

Recall that ║…║ denotes the standard Euclidean norm of the d-dimensional Euclidean space $\mathbb{E}^d$. So, if p _i ; p _j are two points in $\mathbb{E}^d$, then ║p _i p _j ║ denotes the Euclidean distance between them. It is convenient to denote the (finite) point configuration consisting of the points p ₁, p ₂,…, p _N in $\mathbb{E}^d$ by p = (p ₁, p ₂,…p _N ). Now, if p = (p ₁, p ₂,…p _N ) and q = (q ₁, q ₂,…q _N ) are two configurations of N points in $\mathbb{E}^d$ such that for all 1 ≤ i < j ≤ N the inequality ║q _i – q _j ║ ≤ ║p _i – p _j holds, then we say that q is a contraction of p. If q is a contraction of p, then there may or may not be a continuous motion P(t) = (p ₁(t, p ₂(t),…p _N ((t))), with p _i (t) ∈ $\mathbb{E}^d$ for all 0 ≤ t ≤ 1 and 1 ≤ i ≤ N such that p(0) = p and p(1) = q, and ║p _i (t) – p _j (t)║ is monotone decreasing for all 1 ≤ i < j ≤ N. When there is such a motion, we say that q is a continuous contraction of p. Finally, let B ^d[p _i , p _i ] denote the (closed) d-dimensional ball centered at p _i with radius r _i in $\mathbb{E}^d$. In 1954 Poulsen [216] and in 1955 Kneser [183] independently conjectured the following for the case when r ₁ = … = r _N .