The (B)-conjecture for even log-concave measures: log-concavity of t -> mu(e^t K)

Corollary 8.1: for every $n\ge1$, every even log-concave Radon measure $\mu$ on $\mathbb R^n$ (finite or infinite total mass, possibly supported on a subspace) and origin-symmetric convex bodies $K,L$, the log-Brunn-Minkowski inequality holds for $\mu$; taking $L$ a dilate of $K$ gives log-concavity of $t\mapsto\mu(e^tK)$. This is the scalar-dilation form; the stronger diagonal (B)-property for general linear dilations is not claimed except where Saroglou's equivalence for cubes applies.

Corollary 8.1: for every $n\ge1$, every even log-concave Radon measure $\mu$ on $\mathbb R^n$ (finite or infinite total mass, possibly supported on a subspace) and origin-symmetric convex bodies $K,L$, the log-Brunn-Minkowski inequality holds for $\mu$; taking $L$ a dilate of $K$ gives log-concavity of $t\mapsto\mu(e^tK)$. This is the scalar-dilation form; the stronger diagonal (B)-property for general linear dilations is not claimed except where Saroglou's equivalence for cubes applies.