Inequalities for a Circle

Abstract

1. Let | p − q | denote the Euclidean distances between p and q. k. r. Stolarsky proved the following two results:

1° If p₁, p₂, and p₃ are points on the unit circle U, and 0 ≤λ ≤ 2, then there is a p ∈ U such that

$$ \sum\limits_{i = 1}^3 {{{\left| {p - {p_i}} \right|}^i}} \geqslant 2 + {2^\lambda } $$

(1)