A Note on John Simplex with Positive Dilation
This work addresses two open questions regarding Johns theorem for simplices with positive dilation, which is a problem in geometric analysis.
This paper proves a Johns theorem for simplices in R^d with a positive dilation factor of d+2, significantly improving the previous upper bound of d^2. The authors also provide an example demonstrating that d is not the optimal lower bound when d=2.
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}.