The length, width, and inradius of space curves

TL;DR

This paper establishes lower bounds on the length of space curves based on their width and inradius, using topological and integral geometric methods, confirming some conjectures and disproving others.

Contribution

It introduces new bounds on space curve length related to width and inradius, applying advanced geometric techniques and addressing conjectures in the field.

Findings

Confirmed Zalgaller's conjectures up to 99% of their value.

Disproved one of Zalgaller's conjectures.

Provided bounds for both open and closed space curves.

Abstract

The width $w$ of a curve $\gamma$ in Euclidean space $R^n$ is the infimum of the distances between all pairs of parallel hyperplanes which bound $\gamma$, while its inradius $r$ is the supremum of the radii of all spheres which are contained in the convex hull of $\gamma$ and are disjoint from $\gamma$. We use a mixture of topological and integral geometric techniques, including an application of Borsuk Ulam theorem due to Wienholtz and Crofton's formulas, to obtain lower bounds on the length of $\gamma$ subject to constraints on $r$ and $w$. The special case of closed curves is also considered in each category. Our estimates confirm some conjectures of Zalgaller up to $99\%$ of their stated value, while we also disprove one of them.