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  "body": "\n## Definition\n\nLet $n \\ge 1$ and consider [[Euclidean space|0XPR1E85W0]] $\\R^n$.\nA *convex body* in $\\R^n$ is a subset $K \\subseteq \\R^n$ such that:\n\n1. $K$ is [[convex|02AS45T3H9]];\n2. $K$ is [[compact|0R6EWFKRCF]];\n3. $K$ has nonempty [[interior|0RZH6W98HK]].\n\n## Remarks\n\nExample: the closed unit ball $B = \\set{x \\in \\R^n : \\abs{x} \\le 1}$.\nIt is convex by the triangle inequality, compact because it is closed and bounded, and its interior contains the open unit ball, so it is not empty.\n\nNon-example: the closed segment $[0, 1] \\times \\set{0}$ in $\\R^2$ is convex and compact, but its interior in $\\R^2$ is empty, so clause 3 fails.\n\nSome authors, for example in the theory of mixed volumes, call every nonempty compact convex set a convex body and drop clause 3.\nWith that convention, the bodies with nonempty interior are called full-dimensional.\n",
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