Characterizing shapes permitting homothetic bisections

Characterize the sets C in R^d for which every collection of d+1 mass distributions on R^d can be simultaneously bisected by a homothetic copy of C, using only scaling and translation.

Background

The paper establishes simultaneous bisection of d+1 mass distributions by homothetic copies for smooth cookie cutters and by similar copies, allowing rotations and reflections, for arbitrary cookie cutters. The conclusion notes that rotations are needed in the paper only for nonsmooth cookie cutters.

The open problem is to identify precisely which sets retain the universal d+1-mass bisection property when rotations are disallowed and only scaling and translation are permitted.