Large dimension behavior of the Hessian eigenvalues of the unit balls

TL;DR

This paper investigates the asymptotic behavior of Hessian eigenvalues of unit balls in high dimensions, revealing boundedness conditions and growth rates related to the ratio of dimension to eigenvalue order.

Contribution

It establishes bounds and growth rates for k-Hessian eigenvalues in high dimensions and analyzes the limit of Monge–Ampère eigenvalues as dimension increases.

Findings

k-Hessian eigenvalues remain bounded if n/k is bounded

Eigenvalues grow at least as fast as (2-1/k) in n/k ratio

Monge–Ampère eigenvalues tend to 4 as dimension n approaches infinity

Abstract

We show that a sequence of $k$-Hessian eigenvalues of the unit ball in ${\mathbb R}^n$ stays bounded as long as the ratio $n/k$ stays bounded. Moreover, we identify their growth of order at least $(2-1/k)$ in $n/k$. In the case $k=n$, we show that the Monge--Amp\`ere eigenvalues of the unit balls tend to $4$ in the large dimension limit.