The Minkowski problem of $p$-affine dual curvature measures

Abstract

For $p\in (-\infty ,0)\cup (0,1)$ and a convex body $K\subset {\mathbb{R}}^n$ with the origin in its interior, we construct the family of $p$-affine dual curvature measures ${\mathcal{I}}_p(K,\cdot )$ with respect to $K$. The affine-invariant measure ${\tilde{C}}_{n-1}^{\mathrm{a}}(K,\cdot )$ given in the paper [9] is the limit case of ${\mathcal{I}}_p(K,\cdot )$ as $p\to 1^{-}$. The classical cone-volume measure is the limit case of the affine measures $2|p|{\mathcal{I}}_p(K,\cdot )/(n^{2}V({\mathrm{I}}_{p}K))$ when $p\to 0$ and $V(K)=2$, where ${\mathrm{I}}_{p}K$ denotes the $L_p$ intersection body of $K$. The Minkowski problems for the $p$-affine dual curvature measures are proposed and studied. Specifically, we give a sufficient condition for the existence of a solution to the even Minkowski problem for $p$-affine dual curvature measure. Moreover, a necessary condition is given when $p\in (0,1)$. The smooth case of this Minkowski problem is equivalent to solving a new type of partial differential equations with respect to $p$-cosine transforms.