Stability related to the Lp$L_p$ Busemann–Petty problem

Abstract

The Busemann–Petty problem for asks: does imply for origin‐symmetric convex bodies K and L? Here, is the volume of K and the operator is the intersection body of K. In this paper, we study the linear stability and separation results related to the Busemann–Petty problem. A typical result we obtained is: for small enough, if K is an intersection body and L is a star body such that for any , the following inequality: implies where is the Minkowski functional of , and C is a positive number depending only on p and n. Moreover, we also prove the linear stability and separation results for the negative answer of the Busemann–Petty problem.