On one extremal property of a regular simplex

TL;DR

This paper proves that among all simplices of fixed volume, the regular simplex minimizes the Lp-error in asymmetric linear approximation of the quadratic function in d-dimensional space.

Contribution

It establishes a new extremal property of regular simplices related to approximation error minimization for quadratic functions.

Findings

Regular simplices minimize Lp-error for quadratic approximation.

The result applies to simplices of fixed volume in any dimension.

The proof involves geometric and approximation theory techniques.

Abstract

In this paper, we show that the $L_p$-error of asymmetric linear approximation of the quadratic function $Q({\mathbf x})=\sum_{j=1}^{d}x_j^2$ on simplices in $\RR^d$ of fixed volume is minimized on regular simplices.