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  "body": "\n## Definition\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 $\\R^n$.\nFor $j = 0, 1, \\dots, n$, the $j$-th *quermassintegral* of $K$ is\n$$W_j(K) = V(\\underbrace{K, \\dots, K}_{n - j}, \\underbrace{B, \\dots, B}_{j}),$$\nwhere $V$ is the [[mixed volume|QB3Q5FBJ08]].\n\n## Remarks\n\nBy the [[Steiner formula|F02XA3MYX8]], $\\operatorname{vol}(K + tB) = \\sum_{j=0}^n \\binom{n}{j} W_j(K)\\, t^j$ for $t \\ge 0$.\nSpecial values: $W_0(K) = \\operatorname{vol}(K)$, $n W_1(K)$ is the surface area of $K$, and $W_n(K) = \\kappa_n$, the volume of $B$.\nAlso $W_{n-1}(K) = \\frac{\\kappa_n}{2} w(K)$, where $w(K)$ is the mean width of $K$.\nFor $1 \\le k \\le n$, $W_{n-k}(K)$ equals $\\kappa_n / \\kappa_k$ times the average, over all $k$-dimensional linear subspaces $E$, of the $k$-dimensional volume of the orthogonal projection of $K$ onto $E$; this is Kubota's formula.\n\nExample: $K = rB$, a ball of radius $r \\ge 0$.\nBy multilinearity, $W_j(rB) = r^{n-j} V(B, \\dots, B) = \\kappa_n r^{n-j}$.\nFor $n = 2$ and $r = 1$ this gives $W_1 = \\pi$, half the perimeter $2\\pi$, as expected.\n\nNon-example: the set $K = \\set{0, 1} \\subseteq \\R$ is not convex, so its mixed volumes are not defined.\nIndeed the length of $K + tB$ is $4t$ for $t \\le \\frac12$ and $1 + 2t$ for $t \\ge \\frac12$, not one polynomial in $t$.\n",
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