$L^p$-Minkowski Problem under Curvature Pinching

Let $K$ be a smooth, origin-symmetric, strictly convex body in ${\mathbb{R}}^n$. If for some $\mathrm{\ell }\in GL(n,\mathbb{R})$, the anisotropic Riemannian metric $\frac{1}{2}{D}^2\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathrm{\ell }K}^2$, encapsulating the curvature of $\mathrm{\ell }K$, is comparable to the standard Euclidean metric of ${\mathbb{R}}^n$ up-to a factor of $\gamma >1$, we show that $K$ satisfies the even $L^p$-Minkowski inequality and uniqueness in the even $L^p$-Minkowski problem for all $p\ge p_{\gamma }:=1-\frac{n+1}{\gamma }$. This result is sharp as $\gamma \searrow 1$ (characterizing centered ellipsoids in the limit) and improves upon the classical Minkowski inequality for all $\gamma <\mathrm{\infty }$. In particular, whenever $\gamma \le n+1$, the even log-Minkowski inequality and uniqueness in the even log-Minkowski problem hold.