A note on the maximal perimeter and maximal width of a convex small polygon

TL;DR

This paper improves lower bounds for the maximum perimeter and width of convex small polygons with a diameter of one, specifically for cases where the number of sides is a power of two.

Contribution

It advances the understanding of convex small polygons by providing improved lower bounds for maximum perimeter and width when the number of sides is a power of two.

Findings

Enhanced lower bounds for maximum perimeter of small polygons.

Enhanced lower bounds for maximum width of small polygons.

Progress on open problem for polygons with sides n=2^s.

Abstract

The polygon $P$ is small if its diameter equals one. When $n=2^s$, it is still an open problem to find the maximum perimeter or the maximum width of a small $n$-gon. Motivated by Bingane's series of works, we improve the lower bounds for the maximum perimeter and the maximum width.