The complex Busemann–Petty problem on sections of convex bodies

Abstract

The complex Busemann-Petty problem asks whether origin symmetric convex bodies in C n with smaller central hyperplane sections necessarily have smaller volume. We prove that the answer is affirmative if n ≤ 3 and negative if n ≥ 4.

Complex intersection bodies

“…Does it follow that |K| ≤ |L|? As proved in [31], the answer is affirmative if n ≤ 3, and it is negative if n ≥ 4. The proof is based on a connection with intersection bodies, similar to Lutwak's connection in the real case (see [31,Theorem 2]): (i) If K is a complex intersection body in R 2n and L is any origin symmetric complex star body in R 2n , then the answer to the question of the complex Busemann-Petty problem is affirmative; (ii) if there exists an origin symmetric complex convex body in R 2n that is not a complex intersection body , then one can construct a counterexample to the complex Busemann-Petty problem.…”

“…This is no longer true in R 2n , n ≥ 4 as shown in [31,Theorem 4]. The unit balls of complex q -balls with q > 2 are not k-intersection bodies for any 1 ≤ k < 2n − 4.…”

“…The situation with complex convex bodies is different, as no systematic studies of these bodies have been carried out, and results appear only occasionally; see for example [31,35,1,42,49,50].…”

“…Note that the real version of this formula was proved in [23], and that the complex formula below was proved in [31] for infinitely smooth bodies by a different method; here we remove the smoothness condition. Theorem 1.…”

On Polars of Mixed Complex Projection Bodies

“…Until recently the situation with complex convex bodies began to attract attention (see [1,2,9,11,12,13,24,34,36,37]). …”