Identify maximizers of the δ-convolution body volume among equal-volume convex bodies

Determine, for each fixed δ ∈ (0,1), the convex bodies K ⊂ ℝ^n that maximize the n-dimensional volume of the δ-convolution body C_δ K = {x ∈ ℝ^n : vol(K ∩ (K + x)) ≥ δ · vol(K)}, when K ranges over all convex bodies of the same volume.

Background

The paper studies the volume of convolution bodies C_δ K associated with convex bodies K, where the covariogram g_K(x) = vol(K ∩ (K + x)) determines membership via a threshold δ ∈ (0,1). Classical results suggest ellipsoids might be extremal in certain averaged or limiting senses (e.g., via the Petty projection inequality and its extensions), but the authors prove that in the plane ellipsoids do not maximize vol(C_δ K) for any fixed δ.

Despite a negative answer in dimension 2 to the question of whether balls maximize vol(C_δ K) for some δ, the extremal shapes that do maximize vol(C_δ K) for each fixed δ remain unknown. The open problem calls for a characterization or identification of these maximizers under the equal-volume constraint.