Böröczky–Lutwak–Yang–Zhang even $L_p$ Brunn–Minkowski conjecture

Let $L,K$ be origin-symmetric convex bodies in $\mathbb{R}^{n+1}$, and let $h_L,h_K$ be their support functions, while $S_K$ and $V_K$ denote the surface-area and cone-volume measures of $K$. For $p\in[0,1)$, interpret the expressions at $p=0$ by their limit.

Böröczky–Lutwak–Yang–Zhang conjecture. The $L_p$ Brunn–Minkowski inequality holds:

$$V((1-\lambda)L+_p\lambda K)\geq\left((1-\lambda)V(L)^{\frac{p}{n+1}}+\lambda V(K)^{\frac{p}{n+1}}\right)^{\frac{n+1}{p}}.$$

Equivalently,

$$\frac{1}{p}\int_{S^n}h_L^ph_K^{1-p}\,dS_K\geq\frac{n+1}{p}V(K)^{1-\frac{p}{n+1}}V(L)^{\frac{p}{n+1}}.$$

At $p=0$, this is the conjectured log-Minkowski inequality

$$\frac{1}{V(K)}\int_{S^n}\log\frac{h_L}{h_K}\,dV_K\geq\frac{1}{n+1}\log\frac{V(L)}{V(K)}.$$

The conjecture is a central global inequality in the even $L_p$ Brunn–Minkowski theory and is equivalent to the corresponding even $L_p$-Minkowski problem. It is known in dimension $n=1$, while the full range $p\in[0,1)$ remains open.