a mixed volume
Inflate a convex body by adding scaled copies of several other bodies, and ask how its total volume grows. The answer is a polynomial in the scaling amounts, and the coefficients of that polynomial are the mixed volumes — numbers that measure how the bodies interact geometrically. They generalize length, area, and volume into a single graded family: a mixed volume can be 'so much edge of A times so much face of B,' a hybrid measurement blending the boundaries of different bodies.
Precisely, for convex bodies K_1, ..., K_m in R^n and nonnegative scalars lambda_1, ..., lambda_m, Minkowski's theorem says the volume of the dilated combination is a homogeneous polynomial: vol(lambda_1 K_1 + ... + lambda_m K_m) = sum over (i_1,...,i_n) of V(K_{i_1}, ..., K_{i_n}) * lambda_{i_1} ... lambda_{i_n}, where each coefficient V(K_{i_1}, ..., K_{i_n}) is the mixed volume of the n bodies listed (with repetition). They are symmetric in their arguments, multilinear (linear in each slot under Minkowski addition), translation-invariant, and nonnegative; V(K, ..., K) recovers n! times ordinary volume up to normalization, and V(K, ..., K, B) with B the unit ball recovers surface area. Steiner's formula, expanding vol(K + r*B) in powers of r, displays the intrinsic volumes as special mixed volumes.
Mixed volumes are the language in which the deep convex inequalities are stated: Brunn-Minkowski is a statement about a 2-body mixed-volume polynomial, and the Alexandrov-Fenchel inequality is the master inequality relating different mixed volumes. They also appear strikingly outside geometry — the Bernstein-Kushnirenko theorem counts the solutions of a system of polynomial equations by the mixed volume of their Newton polytopes, linking convex geometry to algebraic geometry. An honest caveat: although V is nonnegative and symmetric, it is NOT monotone in the naive sense for all configurations, and mixed volumes can be hard to compute explicitly — even for polytopes the computation is generally #P-hard, so closed forms are special blessings, not the rule.
In R^2 take two convex bodies K, L and expand vol(K + t*L) = vol(K) + 2*V(K, L)*t + vol(L)*t^2. The middle coefficient V(K, L) is the (planar) mixed volume — a mixed area. If L is the unit disk, 2*V(K, disk) equals the perimeter of K, so the mixed volume with a ball literally measures boundary length. For K = L = unit disk this reads area(2*disk) = pi*4 = pi + 2*pi + pi, recovering 4*pi correctly.
Expanding vol(K + tL) in t; the cross-term coefficient is the mixed volume V(K, L).
Mixed volume is a function of n convex bodies at once, not a property of a single body, and it depends on which bodies fill which of the n slots (with repetition). Treating V(K, L) as 'the volume of K times L' is meaningless — it is a coefficient in a dilation polynomial, not a product.