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  "body": "\n## Statement\n\nLet $n \\ge 1$, let $K$ be a nonempty compact convex subset of $\\R^n$ (a [[convex body|437FSG3419]]), and let $B$ be the [[closed unit ball|0Q0H13G8ZB]] of $\\R^n$.\nThen for every $t \\ge 0$,\n$$\\operatorname{vol}(K + tB) = \\sum_{j=0}^{n} \\binom{n}{j} V(K[n - j], B[j]) \\, t^j.$$\nHere $K + tB$ is the [[Minkowski sum|SXS6D87PHE]] and $V$ is the [[mixed volume|QB3Q5FBJ08]].\nThe notation $K[n-j], B[j]$ means $K$ repeated $n - j$ times and $B$ repeated $j$ times.\nIn particular $\\operatorname{vol}(K + tB)$ is a polynomial in $t$ of degree at most $n$.\n\n## Remarks\n\nThe formula is the defining expansion of mixed volumes for $K$ and $B$, with $\\lambda_1 = 1$ and $\\lambda_2 = t$.\nOne groups the terms by the number of copies of $B$.\nThe coefficients $W_j(K) = V(K[n-j], B[j])$ are the [[quermassintegrals|H1K36H9ZWB]].\nRewritten with the [[intrinsic volumes|C2CK6KSSTV]] $V_j$, 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 $\\R^m$.\n\nExample: for a cube $K = [0, a]^3$ in $\\R^3$,\n$$\\operatorname{vol}(K + tB) = a^3 + 6 a^2 t + 3 \\pi a t^2 + \\tfrac{4}{3} \\pi t^3.$$\nThe 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.\n\nConvexity 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 $t \\ge \\frac12$.\n\n## Explanations\n\n### L1 human\n\nThicken a convex shape by a margin $t$ on every side.\nThe volume of the thickened shape is a polynomial in $t$.\nIn the plane it is area plus perimeter times $t$ plus $\\pi t^2$.\nThe coefficients are the basic size measurements of the shape: volume, surface area, and so on down to a constant.\n\n### L2 human\n\nThe Steiner formula is the origin of quermassintegrals and intrinsic volumes.\nIt 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 $K$.\nCombined with the [[Brunn–Minkowski theorem|9B02RPN7XD]] it yields the [[isoperimetric inequality|KBX8RBZ626]].\nIts local versions lead to curvature measures, and a version for sets of positive reach is due to Federer.\n",
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