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

TL;DR

This paper constructs a non-rectifiable simple closed curve in three-dimensional space whose convex hull equals half the Minkowski sum of its range with itself, answering a question about convex hulls of curves.

Contribution

It provides a novel example of a simple closed curve in 3 with a specific convex hull property, addressing an open question in geometric analysis.

Findings

Constructed a non-rectifiable simple closed curve in 3 with convex hull [0,1]^3.

Showed that such a curve cannot be rectifiable.

Established the equality 2(\u03b3)+2(b3)=2(b3) for the constructed curve.

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.