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  "body": "\n## Definition\n\nLet $P \\subset \\R^d$ be a [[convex polytope|300K6CB1KH]] whose [[interior|0RZH6W98HK]] contains the origin $0$.\nLet $x \\cdot y$ denote the [[dot product|0S52TNNTH9]] on $\\R^d$.\nThe *polar polytope* of $P$ is\n$$P^\\circ = \\set{y \\in \\R^d : x \\cdot y \\le 1 \\text{ for all } x \\in P}.$$\n\n## Remarks\n\nIf $v_1, \\dots, v_m$ are the vertices of $P$, then $P^\\circ = \\set{y : v_i \\cdot y \\le 1, \\ i = 1, \\dots, m}$.\n$P^\\circ$ is again a convex polytope with $0$ in its interior, and $(P^\\circ)^\\circ = P$.\nThe map $F \\mapsto \\set{y \\in P^\\circ : x \\cdot y = 1 \\text{ for all } x \\in F}$ is an inclusion-reversing bijection from the [[faces|KY7SB854VH]] of $P$ to the faces of $P^\\circ$, sending $k$-faces to $(d - 1 - k)$-faces.\n\nExample: let $P = [-1, 1]^2$.\nIts vertices are $(\\pm 1, \\pm 1)$, so $y \\in P^\\circ$ means $\\pm y_1 \\pm y_2 \\le 1$ for all signs, that is $\\abs{y_1} + \\abs{y_2} \\le 1$.\nSo $P^\\circ$ is the square with vertices $(\\pm 1, 0)$ and $(0, \\pm 1)$.\n\nNon-example: for $P = [0,1]^2$ the origin is a vertex, not an interior point, so the definition does not apply.\nThe same formula gives the unbounded set $\\set{y : y_1 \\le 1, \\ y_2 \\le 1, \\ y_1 + y_2 \\le 1}$, which is not a polytope.\n",
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