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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $K$ be a nonempty compact convex subset of $\\R^n$ (a [[convex body|437FSG3419]]).\nFor $m \\ge 0$ let $\\kappa_m$ be the volume of the closed unit ball of $\\R^m$, so $\\kappa_0 = 1$, $\\kappa_1 = 2$, $\\kappa_2 = \\pi$.\nFor $j = 0, 1, \\dots, n$, the $j$-th *intrinsic volume* of $K$ is\n$$V_j(K) = \\binom{n}{j} \\frac{W_{n-j}(K)}{\\kappa_{n-j}},$$\nwhere $W_{n-j}$ is the $(n-j)$-th [[quermassintegral|H1K36H9ZWB]].\n\n## Remarks\n\nEquivalently, with $B$ the closed unit ball, the Steiner formula reads\n$$\\operatorname{vol}(K + tB) = \\sum_{j=0}^n \\kappa_{n-j} V_j(K)\\, t^{n-j} \\quad (t \\ge 0).$$\nSpecial values: $V_n(K) = \\operatorname{vol}(K)$, $V_{n-1}(K)$ is half the surface area, and $V_0(K) = 1$.\nIf $K$ lies in a $k$-dimensional affine subspace, then $V_j(K)$ is the same whether computed in $\\R^n$ or in that subspace, and $V_j(K) = 0$ for $j \u003e k$.\n\nExample: the unit square $K = [0,1]^2$ in $\\R^2$.\nIts parallel body has area $1 + 4t + \\pi t^2$, and the formula above gives $V_2(K) + 2 V_1(K)\\, t + \\pi V_0(K)\\, t^2$.\nSo $V_2(K) = 1$, $V_1(K) = 2$ (half the perimeter) and $V_0(K) = 1$.\n\nNon-example: $K = \\set{0, 1} \\subseteq \\R$ is compact but not convex.\nThe length of $K + tB$ is $4t$ for $t \\le \\frac12$ and $1 + 2t$ for $t \\ge \\frac12$, which is not one polynomial, so there are no intrinsic volumes of $K$ in this sense.\n",
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