the Alexandrov-Fenchel inequality
/ uh-leck-SAN-drof FEN-khel /
Mixed volumes are coefficients measuring how convex bodies blend; they obey a hidden curvature law. The Alexandrov-Fenchel inequality is the master statement of that law: a single quadratic inequality among mixed volumes that says, roughly, 'the cross term cannot exceed the geometric mean of the two diagonal terms.' Nearly every classical inequality of convex geometry — Brunn-Minkowski, the isoperimetric inequality, the inequalities between quermassintegrals — is a special case obtained by choosing the bodies cleverly.
Precisely, for convex bodies K, L, and a list C = (C_1, ..., C_{n-2}) of n - 2 further convex bodies in R^n, the mixed volume V(K, L, C) satisfies V(K, L, C)^2 >= V(K, K, C) * V(L, L, C). In words, the mixed volume is a 'log-concave' bilinear form in its first two slots: the matrix with these three entries is negative-semidefinite on the relevant subspace, a quadratic-form inequality. Setting all the C_i equal to the unit ball reduces it to the classical Minkowski quadratic inequalities between intrinsic volumes; setting things up with K + L recovers Brunn-Minkowski. The inequality is genuinely deep — Alexandrov gave two proofs in the 1930s, and the equality cases were only fully resolved much later.
Its reach is enormous and surprising. Inside geometry it unifies the isoperimetric-type inequalities and underlies the theory of valuations. Outside it has spectacular applications: Stanley used a combinatorial Alexandrov-Fenchel inequality to prove monotonicity results about linear extensions of partially ordered sets (the celebrated XYZ-type and log-concavity theorems), connecting convex geometry to combinatorics and even to the resolution of long-standing counting conjectures. Two honest caveats. First, the equality cases are notoriously subtle — far harder than the inequality itself, and a complete classification was a major modern achievement. Second, despite its power, the inequality is non-trivial to apply: choosing the auxiliary bodies C to extract a desired classical inequality is an art, not a formula.
In the plane (n = 2) there are no extra bodies C, and the inequality reads V(K, L)^2 >= V(K, K) * V(L, L) = area(K) * area(L). Taking L the unit disk, V(K, disk) is half the perimeter of K, so this becomes (perimeter/2)^2 >= area(K) * pi, i.e. perimeter^2 >= 4*pi*area — exactly the isoperimetric inequality, with equality only for a disk. Alexandrov-Fenchel hands you isoperimetry as a one-line corollary.
In the plane the inequality collapses to perimeter^2 >= 4*pi*area — the isoperimetric inequality.
The equality cases of Alexandrov-Fenchel are dramatically harder than the inequality and were not fully understood for decades; do not assume the naive 'K and L homothetic' characterization always holds — in higher mixed settings degenerate equality configurations occur.