Stability of volume comparison for complex convex bodies

Abstract

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 ${{\mathbb C}^n}$, n = 2 or n = 3, ${\varepsilon >0 }$ and ${{\rm Vol}_{2n-2}(K\cap H) \le {\rm Vol}_{2n-2}(L \cap H) + \varepsilon}$ for any complex hyperplane H in ${{\mathbb C}^n}$ , then ${({\rm Vol}_{2n}(K))^{\frac{n-1}n}\le({\rm Vol}_{2n}(L))^{\frac{n-1}n} + \varepsilon}$ , where Vol_2n is the volume in ${{\mathbb C}^n}$ , which is identified with ${{\mathbb R}^{2n}}$ in the natural way.