An Isodiametric Problem with Additional Constraints in Euclidean space R^3

TL;DR

This paper investigates the maximal volume convex domain within a circular cone in three-dimensional space, establishing its uniqueness and shape under specific constraints.

Contribution

It introduces a new isodiametric problem with additional geometric constraints and determines the unique maximal volume convex domain in this setting.

Findings

Existence of a unique maximal volume convex domain.

Explicit determination of the shape of the maximal domain.

Proof of the domain's inclusion within the cone with specified properties.

Abstract

Let $C_{\theta}$ be a circular cone in Euclidean space $\mathbb{R}^{3}$,which apex is the origin and apex angle of the cone is $\theta\in \left(\pi/3, \pi\right)$. Let $M_\theta$ be the class of compact convex domains in Euclidean space $\mathbb{R}^{3}$, which have diameter one, contains the origin and are included in $C_{\theta}$. In this paper, we show that there is a unique compact convex domain with maximal volume and also we determine the shape of the above domain.