The Affine Convex Lorentz–Sobolev Inequalities

Stability of the Borell–Brascamp–Lieb Inequality for Multiple Power Concave Functions

“…Moreover, the equality in (1) holds if and only if K 1 and K 2 are homothetic, i.e., K 1 = sK 2 + x, for some s > 0 and x ∈ R n . The Brunn-Minkowski inequality is one of the fundamental results in the theory of convex bodies, and several other important inequalities, e.g., the isoperimetric inequality, can be deduced from it; see [1][2][3][4][5][6], for example. Stability results of an inequality answer the following question: is the inequality that we consider sensitive to small perturbations of the maximizers (or minimizers) of the inequality?…”

“…In the case m = 2, it turns to the classical relative asymmetry (2). Note that it is essentially the minimum of A(K i , K j ), that is,…”