  We interpret the log-Brunn-Minkowski conjecture of
B\"or\"oczky-Lutwak-Yang-Zhang as a spectral problem in centro-affine
differential geometry. In particular, we show that the Hilbert-Brunn-Minkowski
operator coincides with the centro-affine Laplacian, thus obtaining a new
avenue for tackling the conjecture using insights from affine differential
geometry. As every strongly convex hypersurface in $\mathbb{R}^n$ is a
centro-affine unit-sphere (having constant centro-affine Ricci curvature equal
to $n-2$), we may use the classical argument of Lichnerowicz and a
centro-affine Bochner formula to give a new proof of the Brunn-Minkowski
inequality. For origin-symmetric convex bodies enjoying fairly generous
curvature pinching bounds (improving with dimension), we are able to show
global uniqueness in the $L^p$- and log-Minkowski problems, as well as the
corresponding global $L^p$- and log-Minkowski conjectured inequalities. As a
consequence, we resolve the isomorphic version of the log-Minkowski problem:
for any origin-symmetric convex body $\bar K$ in $\mathbb{R}^n$, there exists
an origin-symmetric convex body $K$ with $\bar K \subset K \subset 8 \bar K$,
so that $K$ satisfies the log-Minkowski conjectured inequality, and so that $K$
is uniquely determined by its cone-volume measure $V_K$. Analogous isometric
results are derived as well.
