$L_p$-Integral Affine Surface Area-Variational Capacity Inequalities

Abstract

For $n\ge 2$ and $1<p<n$, Ludwig-Xiao-Zhang \cite{LXYZ} asked whether the $L_p$-integral affine surface area of every convex body containing the origin in its interior is bounded above by the ball-normalized multiple of \(L_p\)-variational capacity, and whether the analogous analytic inequality holds for $L_p$-convexified level sets. We answer both questions negatively.