Continuity of the solution to the Minkowski problem for $L_p$ torsional measure

Abstract

This paper deals with the continuity of the solution to the Minkowski problem for $L_p$ torsional measure. For $p\in(1,n+2)\cup (n+2,\infty)$, we show that a sequence of convex bodies in $\mathbb{R}^n$ is convergent in Hausdorff metric if the sequence of the $L_p$ torsional measures (associated with these convex bodies) is weakly convergent. Moreover, we also prove that the solution to the Minkowski problem for $L_p$ torsional measure is continuous with respect to $p$.