The Maximum of the Volume of a Part of a Cevian Simplex

TL;DR

This paper investigates the maximum ratio of the combined volume of certain smaller simplices formed by cevians passing through a point within a larger simplex, with a specific example provided for tetrahedra.

Contribution

It introduces a new problem of maximizing volume ratios in cevian simplices and provides insights into the geometric configurations involved.

Findings

Derived bounds for volume ratios in cevian simplices

Analyzed the case of tetrahedra as a specific example

Proposed methods for maximizing volume sums within simplices

Abstract

The cevians passing through a point in a simplex create a cevian simplex, which is divided by these cevians into smaller simplices. We consider the problem about the maximum of the ratio of the sum of the volumes of some of these smaller simplices by the volume of the reference simplex. The special case of tetrahedron is given as an example.