Quermassintegrals

Quermassintegrals, also known as intrinsic volumes in convex geometry, are a finite sequence of functionals W0(K), W1(K), ..., Wn(K) defined for a convex body K in n-dimensional Euclidean space. They arise naturally in Steiner’s formula, which describes how the volume of K expands when K is dilated by a ball. If B denotes the unit ball and κm its volume, then for t ≥ 0 the volume of K dilated by tB can be written as Vol(K + tB) = ∑i=0n κn−i Wi(K) ti. The numbers Wi(K) are the quermassintegrals.

The quermassintegrals form a hierarchy of geometric measurements. W0(K) equals Vol(K). In the plane (n = 2), the classical example is Vol(K + tB) = Area(K) + Perimeter(K) t + π t^2, which shows W0(K) = Area(K), W1(K) = Perimeter(K)/2, and W2(K) = π. In higher dimensions, the Wi(K) generalize volume, surface area, mean curvature integrals, and other curvature-related quantities. They are homogeneous of degree i under scaling (Wi(λK) = λi Wi(K)) and are monotone with respect to inclusion.

Quermassintegrals are intrinsically tied to Minkowski functionals and mixed volumes, providing a unified framework for many classical geometric invariants. They satisfy fundamental inequalities and play a central role in integral geometry, geometric inequalities, and affine and convex-geometry theory. The concept originates from Steiner’s formula and has been developed under various names, including intrinsic volumes and quermassintegrale.