A Simple Closed Curve in \(\mathbb{R}^3\) Whose Convex Hull Equals the Half-sum of the Curve with Itself

Abstract

If \(\Gamma\) is the range of a Jordan curve that bounds a convex set in the plane, then \(\frac{1}{2}(\Gamma+\Gamma)=\mathsf{co}(\Gamma),\) where \(+\) is the Minkowski sum and \(\mathsf{co}\) is the convex hull. Answering a question of V. N. Ushakov, we construct a simple closed curve in \(\mathbb{R}^3\) whose range \(\Gamma\) satisfies \(\frac{1}{2}(\Gamma+\Gamma)=\mathsf{co}(\Gamma)=[0,1]^3.\) Also we show that such a simple closed curve cannot be rectifiable.