Stability of volume comparison for complex convex bodies

摘要

We prove the stability of the affirmative part of the solution to the complex Busemann-Petty problem. Namely, if K and L are origin-symmetric convex bodies in ?~n, n = 2 or n = 3, e{open} > 0 and Vol_(2n-2)(K ∩ H) ≤ Vol_(2n-2)(L ∩ H) + e{open} for any complex hyperplane H in ?~n, then is the volume in ?~n, which is identified with ?~(2n) in the natural way.