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 Rⁿ 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--Ampère eigenvalues of the unit balls tend to $4$ in the large dimension limit.