# Torsion of locally convex curves

## TL;DR

This paper proves that the torsion of certain star-shaped, locally convex space curves must change sign at least four times, extending the classical four vertex theorem to a broader class of curves.

## Contribution

It generalizes Sedykh's four vertex theorem to star-shaped, locally convex curves in space, using a reduction to spherical curve inflections and Arnold's tennis ball theorem.

## Key findings

- Torsion changes sign at least four times for the specified curves.
- The result extends classical convex curve theorems to a broader class.
- Reduction to spherical curve inflections links space curve torsion to known theorems.

## Abstract

We show that the torsion of any simple closed curve $\Gamma$ in Euclidean 3-space changes sign at least $4$ times provided that it is star-shaped and locally convex with respect to a point $o$ in the interior of its convex hull. The latter condition means that through each point $p$ of $\Gamma$ there passes a plane $H$, not containing $o$, such that a neighborhood of $p$ in $\Gamma$ lies on the same side of $H$ as does $o$. This generalizes the four vertex theorem of Sedykh for convex space curves. Following Thorbergsson and Umehara, we reduce the proof to the result of Segre on inflections of spherical curves, which is also known as Arnold's tennis ball theorem.