Simplex and cube extremality conjecture for the Monge–Ampère eigenvalue

Let $n\geq 2$. A bounded open convex set in $\mathbb R^n$ has fixed positive volume, and the Monge–Ampère eigenvalue is the principal eigenvalue associated with the Monge–Ampère operator on that set. In the centrally symmetric case, assume the set is open, bounded, convex, and centrally symmetric.

Simplex and cube extremality conjecture. (i) Among all bounded open convex sets in $\mathbb R^n$ having a fixed positive volume, the $n$-dimensional regular simplex—the interior of the convex hull of $n+1$ equally spaced points in $\mathbb R^n$—has the smallest Monge–Ampère eigenvalue. (ii) Among all open bounded centrally symmetric convex sets in $\mathbb R^n$ having a fixed positive volume, the $n$-dimensional cube has the smallest Monge–Ampère eigenvalue.

The conjecture identifies the proposed minimizers in the extremal inequalities for the Monge–Ampère eigenvalue, motivated by Ball's reverse isoperimetric inequality. The cited context indicates that determining the extremal convex sets remains an open problem, including both the general and centrally symmetric cases.