The logarithmic Brunn–Minkowski conjecture

We prove the logarithmic Brunn–Minkowski conjecture for arbitrary origin-symmetric convex bodies in every dimension. The theorem also gives the symmetric L${}_p$ Brunn–Minkowski inequality for every $0<p<1$. Combined with Saroglou's transfer theorem and a support-subspace reduction, it yields the logarithmic inequality for every even log-concave Radon measure and the scalar-dilation $(B)$-conjecture.