A New Approach to the Minkowski First Mixed Volume and the LYZ Conjecture

The variation of the functional U of a convex body in \({\mathbb {R}}^n\) introduced by Lutwak–Yang–Zhang is derived. It becomes the first mixed volume of Minkowski when the convex body is strictly convex. A Minkowski-type inequality for the variation of the of U is proved, which implies the LYZ conjecture for the functional U directly.