A very short unified proof of Dar's conjecture and Log-Brunn--Minkowski in the plane

Abstract

This short note gives a very short unified proof of both Dar's conjecture and the log-Brunn--Minkowski inequality in the plane. A planar $L_p$-Brunn-Minkowski inequality for $p\in [-\infty ,0]$ is established and gives the two key inequalities at $p=0$ and $p=-\infty$. Motivated by this new proof, the author proposes a related conjecture in $\tr^3$, to attack the two conjectures in higher dimensions.