Fundamental Gaps for the Dirichlet p-Laplacian on Convex Domains

We study the fundamental gap of the Dirichlet $p$-Laplacian on bounded convex domains. We first prove the sharp one-dimensional estimate $Γ_p(I_D,V)\ge (p-1)(2^p-1)(π_p/D)^p$, $I_D=(-D/2,D/2)$, for all $1<p<\infty$ and convex potentials $V$, with equality exactly for constant $V$. An analysis of col