The Maximum of the Volume of a Cevian Simplex and its Parts

TL;DR

This paper generalizes the maximum volume of cevian triangles in a triangle to n-dimensional simplices, establishing bounds and solving related optimization problems using barycentric coordinates.

Contribution

It extends the known area bound for cevian triangles to higher dimensions and applies barycentric coordinates to solve optimization problems.

Findings

Maximum volume of cevian simplex is rac{1}{n^n} of the original simplex.

The maximum is attained when the interior point is the centroid.

Provides solutions to two optimization problems related to simplex parts.

Abstract

The cevian triangle corresponding to an interior point $M$ of a triangle is the triangle determined by the feet of the three cevians concurrent at $M$. It is known that the area of the cevian triangle for an interior point $M$ of a triangle is at most $\frac{1}{4}$ of the area of the triangle, with maximum attained when $M$ is the triangle's centroid. This can be generalized from triangles to $n$-dimensional simplices, with $\frac{1}{4}$ replaced by $\frac{1}{n^n}$, using barycentric coordinates. We also use this method to solve two optimization problems about the parts of this simplex.