Steiner formula

Remarks

The formula is the defining expansion of mixed volumes for $K$ and $B$, with $\lambda_1 = 1$ and $\lambda_2 = t$. One groups the terms by the number of copies of $B$. The coefficients $W_j(K) = V(K[n-j], B[j])$ are the quermassintegrals. Rewritten with the intrinsic volumes $V_j$, it reads $\operatorname{vol}(K + tB) = \sum_{j=0}^n \kappa_{n-j} V_j(K)\, t^{n-j}$, where $\kappa_m$ is the volume of the unit ball of $\R^m$.

Example: for a cube $K = [0, a]^3$ in $\R^3$, $$\operatorname{vol}(K + tB) = a^3 + 6 a^2 t + 3 \pi a t^2 + \tfrac{4}{3} \pi t^3.$$ The terms come from the cube, six slabs over the faces, twelve quarter cylinders along the edges, and eight eighths of a ball at the corners.

Convexity is needed for a polynomial: for $K = \set{0, 1} \subseteq \R$ the length of $K + tB$ is $4t$ for $t \le \frac12$ and $1 + 2t$ for $t \ge \frac12$.

Going deeper

For a student

The Steiner formula is the origin of quermassintegrals and intrinsic volumes. It explains why surface area is a derivative of volume: $\frac{d}{dt} \operatorname{vol}(K + tB)$ at $t = 0$ is $n V(K[n-1], B)$, the surface area of $K$. Combined with the Brunn–Minkowski theorem it yields the isoperimetric inequality. Its local versions lead to curvature measures, and a version for sets of positive reach is due to Federer.