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 $n/k$ 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.