Several problems on reduced spherical polygons of thickness less than {\pi}/2

TL;DR

This paper investigates properties of reduced spherical polygons with thickness less than π/2, establishing minimal perimeter, maximal diameter, and minimal containing radius, thus advancing geometric understanding of these shapes.

Contribution

It proves that regular spherical n-gons minimize perimeter and determines bounds on diameter and radius for reduced spherical polygons with fixed thickness.

Findings

Regular spherical n-gons have minimal perimeter among their class.

Maximal diameter is characterized for polygons with fixed thickness.

Smallest spherical radius containing all such polygons is identified.

Abstract

The present paper aims to solve some problems proposed by Lassak about the reduced spherical polygons. The main result is to show that the regular spherical n-gon has the minimal perimeter among all reduced spherical polygons of fixed thickness less than {\pi}/2 and with at most n vertices. In addition, we determine the maximal diameter of every reduced spherical polygons with a fixed thickness less than {\pi}/2. We also find the smallest spherical radius that contains every reduced spherical polygons with a fixed thickness less than {\pi}/2.