A Simple Closed Curve in ℝ³ Whose Convex Hull Equals the Half-Sum of the Curve with Itself

Abstract

If Γ is the range of a Jordan curve that bounds a convex set in R 2 , then 1 2 (Γ + Γ) = co(Γ), where + is the Minkowski sum and co is the convex hull. Answering a question of V.N. Ushakov, we construct a simple closed curve in R 3 with range Γ such that 1 2 (Γ + Γ) = [0, 1] 3 = co(Γ). Also we show that such simple closed curve cannot be rectifiable.