Böröczky–Lutwak–Yang–Zhang log-Brunn–Minkowski inequality

Let $K,L$ be symmetric convex bodies in $\mathbb R^n$. For $\lambda\theta \in [0,1]$, define their logarithmic Minkowski combination by

$$(1-\lambda) \cdot K \oplus_0 \lambda \cdot L = \{ x \in \mathbb R^n: \langle x,u \rangle \leq h_K(u)^{1-\lambda} h_L(u)^{\lambda} \text{ for all } u\in S^{n-1} \},$$

where $h_K$ and $h_L$ are support functions and $|\cdot|$ denotes Lebesgue measure. Böröczky–Lutwak–Yang–Zhang's log-Brunn–Minkowski inequality. For symmetric convex bodies $K,L$ in $\mathbb R^n$ and $\lambda \in [0,1]$,

$$|(1-\lambda) \cdot K \oplus_0 \lambda \cdot L| \geq |K|^{1-\lambda}|L|^\lambda.$$

Böröczky et al. proved the inequality in the plane, and Saroglou proved it for unconditional convex bodies; its validity for arbitrary symmetric convex bodies in general dimension is not established here and remains open.