A simple closed curve in ${\mathbb{R}}^3$ whose convex hull equals the half-sum of the curve with itself

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