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 $\mathbb{R}^2,$ 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 $\mathbb{R}^3$ with range $\Gamma$ such that $\frac{1}{2}(\Gamma+\Gamma)=[0,1]^3=\mathsf{co}(\Gamma).$ Also we show that such simple closed curve cannot be rectifiable.