Center-convexity for general convex bodies (conjecture)

Prove that for any convex body K ⊂ R^n, the self-perimeter functional p ↦ P(K−{p}) is (strictly) convex on Int(K), where P denotes the self-perimeter defined by P(B) = ∫_{∂*B} [ V(B^{(n−1)}(ν_x)) / 𝓗_{n−1}(B^{(n−1)}(ν_x)) ] d𝓗_{n−1}(x). This would extend the established center-convexity for Cartesian products of simplexes and two-dimensional convex sets to all convex bodies.

Background

The authors prove strict convexity of the self-perimeter functional with respect to the center for certain classes, including n-simplexes and Cartesian products of simplexes and 2D convex sets, ensuring a unique minimizing point. This mirrors the known two-dimensional result that the self-perimeter is strictly convex in the interior point.

They state a conjecture that this center-convexity property should hold for general convex bodies in Rn. Establishing this would imply the uniqueness of an optimal center minimizing self-perimeter for arbitrary convex bodies in higher dimensions.

References

This implies the center-convexity of any convex body in $n$ composed of Cartesian product of simplexes and $2-D$ convex sets. The center convexity of general convex body is conjectured,

— Self perimeter of convex sets  (2604.01950 - Wolansky, 2 Apr 2026) in Subsection “Extension to higer dimensions” in the Introduction