For origin-symmetric convex bodies, the dual Brunn-Minkowski inequality holds for all orders 0<q≤n, and the cube maximizes dual quermassintegrals among bodies in John's position.
Kolesnikov and Emanuel Milman,Brascamp-Lieb-type inequalities on weighted Riemannian manifolds with boundary, J
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Brunn-Minkowski and Reverse Isoperimetric Inequalities for Dual Quermassintegrals
For origin-symmetric convex bodies, the dual Brunn-Minkowski inequality holds for all orders 0<q≤n, and the cube maximizes dual quermassintegrals among bodies in John's position.