Alexandrov–Fenchel inequality

Remarks

The inequality was proved by Alexandrov, and independently by Fenchel, in the 1930s. Alexandrov's proofs treat polytopes or smooth bodies first, by induction on the dimension, and then pass to general bodies by approximation. Khovanskii and Teissier found a proof through the Hodge index theorem applied to toric varieties.

Example: for $n = 2$ the inequality reads $V(K, L)^2 \ge \operatorname{area}(K) \operatorname{area}(L)$. With $L$ the unit disk, $V(K, L) = \frac12 \operatorname{per}(K)$, and one gets $\operatorname{per}(K)^2 \ge 4 \pi \operatorname{area}(K)$, the planar isoperimetric inequality.

Going deeper

For a student

Applied with the remaining arguments chosen among $K_1$ and $K_2$, it shows that the sequence $V(K_1[i], K_2[n - i])$, for $i = 0, \dots, n$, is log-concave in $i$. It implies Minkowski's first inequality and the Brunn–Minkowski theorem for convex bodies. Stanley used it to prove that the numbers of linear extensions of a poset with a fixed element at a given position form a log-concave sequence. Its equality cases are understood only in special situations, and their full description is a hard problem.