Equality cases in McMullen’s fiber Brunn–Minkowski inequality

Determine the necessary and sufficient conditions for equality in McMullen’s Brunn–Minkowski-type inequality for fiber combinations: for complementary subspaces L and M of R^n with ℓ = dim L, convex bodies K1, K2 ⊂ R^n, and θ ∈ [0,1], the inequality asserts that the n-dimensional volume of the fiber combination of the fiber-dilates (1−θ)∘_{L|M}K1 and θ∘_{L|M}K2, raised to the power 1/ℓ, is at least (1−θ)·vol_n(K1)^{1/ℓ} + θ·vol_n(K2)^{1/ℓ}; characterize precisely when equality holds.

Background

McMullen introduced the fiber combination of convex bodies relative to complementary subspaces L and M, where vectors are added in their L-components while their M-components are fixed. This operation interpolates between Minkowski addition and intersection and yields the Steiner symmetral when L is a line and M is its orthogonal hyperplane.

He proved a Brunn–Minkowski-type inequality for the fiber combination: setting ℓ = dim L, the functional vol_n(·)^{1/ℓ} is concave with respect to fiber addition of appropriately dilated bodies. This establishes an analogue of the Brunn–Minkowski inequality in the fiber setting.

However, the equality cases for this fiber Brunn–Minkowski inequality have not been identified. Establishing a full characterization would parallel the classical equality conditions in the standard Brunn–Minkowski theory and clarify the geometric structure of extremizers for fiber combinations.