A Note on John Simplex with Positive Dilation

TL;DR

This paper proves a tighter upper bound for Johns theorem on simplices with positive dilation in Euclidean space, improving previous bounds and providing new insights into optimal bounds.

Contribution

It establishes a new upper bound of d+2 for simplices with positive dilation, improving the previous d^2 bound, and explores the bounds' tightness.

Findings

Upper bound of d+2 for simplices with positive dilation

The bound is tight based on the d lower bound

Counterexample showing d isn't the optimal lower bound when d=2

Abstract

We prove a Johns theorem for simplices in $R^d$ with positive dilation factor $d+2$, which improves the previously known $d^2$ upper bound. This bound is tight in view of the $d$ lower bound. Furthermore, we give an example that $d$ isn't the optimal lower bound when $d=2$. Our results answered both questions regarding Johns theorem for simplices with positive dilation raised by \cite{leme2020costly}.