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.