Stability of the Borell–Brascamp–Lieb inequality under near-equality

Determine whether near-equality in the Borell–Brascamp–Lieb inequality h((1 − λ)y + λz) ≥ M_α(f(y), g(z); λ) with the integral inequality ∫ h ≥ M_{α/(1+nα)}(∫ f, ∫ g; λ) implies that the functions f, g, and h are close (in an appropriate quantitative sense) to satisfying the equality characterization of Theorem 1.2; specifically, ascertain whether they must be close to being α-concave and homothetic to each other.

Background

The paper recalls that rigidity and equality conditions for the Borell–Brascamp–Lieb inequality (BBL) are known (Dubuc’s theorem), and then poses the natural next step: stability. The authors formulate the concrete question of whether near-equality forces the participating functions to be approximately α-concave and approximately homothetic, emphasizing that these are distinct aspects.

They note that this stability problem is largely open, with only limited results available and few that address unrestricted classes of functions. The relevance extends to the stability of the Brunn–Minkowski inequality, which is more developed but still has recent breakthroughs beyond convex sets.