Extremal self-volume in higher dimensions

Determine the supremum and the infimum of the self-volume V_B^{(n)} among all centrally symmetric convex bodies B ⊂ R^n for dimensions n > 2, where the self-volume is defined by V_B^{(n)} := P(B)/n and the self-perimeter P(B) is given by the boundary integral P(B) = ∫_{∂*B} [ V(B^{(n−1)}(ν_x)) / 𝓗_{n−1}(B^{(n−1)}(ν_x)) ] d𝓗_{n−1}(x), with B^{(n−1)}(ν_x) = B ∩ Σ(ν_x).

Background

The paper introduces a recursive definition of self-perimeter and self-volume for convex sets in R^n that preserves the Euclidean ratio P(B)/V(B) = n and is invariant under origin-preserving affine transformations. For n = 2, many classical estimates are known; however, higher-dimensional behavior is largely unexplored.

Within this framework, the authors explicitly pose open questions regarding the extremal values of the self-volume V_B^{(n)} among centrally symmetric convex sets for n > 2, highlighting the lack of known bounds or extremal shapes in higher dimensions.