Determine equality conditions for fractional subadditivity in dimensions n ≥ 2

Determine the equality conditions, for n ≥ 2, in the fractional subadditivity inequality for the Schneider non-convexity index c: for any m-tuple of non-empty compact sets A1, …, Am in R^n and any fractional partition (G, β) of [m], characterize exactly when equality holds in c(A1 + … + Am) ≤ ∑_{S∈G} β(S) c(∑_{i∈S} Ai). Provide a higher-dimensional analogue of the complete one-dimensional equality characterization established in the paper.

Background

The authors prove that the Schneider non-convexity index is fractionally subadditive for arbitrary m and n, and they give a complete characterization of equality conditions when n = 1. This uses a translation-invariance reduction and specific structural arguments in one dimension.

However, in higher dimensions, Minkowski addition with a convex set does not necessarily strictly reduce the Schneider index (a counterexample is provided), preventing a straightforward extension of the one-dimensional equality analysis.