Modified Busemann-Petty problem on sections of convex bodies

Abstract

The Busemann-Petty problem asks whether originsymmetric convex bodies in R n with smaller central hyperplane sections necessarily have smaller n-dimensional volume. It is known that the answer is affirmative if n ≤ 4 and negative if n ≥ 5. In this article we modify the assumptions of the original Busemann-Petty problem to guarantee the affirmative answer in all dimensions.

The modified complex Busemann-Petty problem on sections of convex bodies

“…Since the answer is negative in most dimensions, it is natural to ask what conditions on the (n − 1)-dimensional volumes of central sections do allow to compare the n-dimensional volumes. Such conditions were found in [16]. The result is as follows.…”

“…In this article we give necessary conditions on the section function in order to obtain an affirmative answer in all dimensions. The result is the complex analogue of [16]. …”

“…The construction of the body follows similar steps as in [16]. We put q = 2n − α − 4, so q ∈ (2, 3).…”

Comparison of volumes of convex bodies in real, complex, and quaternionic spaces

“…An important feature of these operators is that the corresponding Fourier multiplier |y| α does not preserve the Schwartz space S(R N ) and the phrases like "in the sense of distributions" (cf. [36,35,74]) require careful explanation and justification. Section 4 is devoted to weighted section functions of origin-symmetric convex bodies.…”

“…It is "Yes" if and only if n 4; see [15,16,33,35], and references therein. The second question, related to implication (1.1), was asked by Koldobsky, Yaskin and Yaskina [36]. It was called the modified Busemann-Petty problem.…”