L ^p -Minkowski Problem Under Curvature Pinching

Let be a smooth, origin-symmetric, strictly convex body in. If for some, the anisotropic Riemannian metric, encapsulating the curvature of, is comparable to the standard Euclidean metric of up-to a factor of 1$]]>, we show that satisfies the even -Minkowski inequality and uniqueness in the even -Minkowski problem for all. This result is sharp as (characterizing centered ellipsoids in the limit) and improves upon the classical Minkowski inequality for all <![CDATA[$\gamma. In particular, whenever, the even log-Minkowski inequality and uniqueness in the even log-Minkowski problem hold.