  Let $K$ be a smooth, origin-symmetric, strictly convex body in
$\mathbb{R}^n$. If for some $\ell\in GL(n,\mathbb{R})$, the anisotropic
Riemannian metric $\frac{1}{2}D^2 \Vert\cdot\Vert_{\ell K}^2$, encapsulating
the curvature of $\ell K$, 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 $p \geq p_\gamma := 1 - \frac{n+1}{\gamma}$. This result is
sharp as $\gamma \searrow 1$ (characterizing centered ellipsoids in the limit)
and improves upon the classical Minkowski inequality for all $\gamma < \infty$.
In particular, whenever $\gamma \leq n+1$, the even log-Minkowski inequality
and uniqueness in the even log-Minkowski problem hold.
