Uniqueness for the Degenerate Monge-Ampère Equation on Arbitrary Bounded Convex Domains

Let \(n\ge 2\) and let \(\Omega \subset \mathbb R^n\) be an arbitrary bounded open convex set. We prove that, for \(p>n\), the Dirichlet problem

has at most one convex Alexandrov solution. The proof is based on the affine behavior of the Monge-Ampère energy and on a power-concavity property of the \(L^{p+1}\) mass, along the Legendre path connecting two solutions. At the homogeneous exponent \(p=n\), the same argument shows that any two nonzero solutions with the same coefficient are positive multiples of one another.