Large dimension behavior of the Hessian eigenvalues of the unit balls

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