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  "body": "\n## Definition\n\nLet $V$ be a real [[vector space|000AYPA9AJ]], let $I$ be a set, and let $(x_i)_{i \\in I}$ be a family of points of $V$.\nThe family is *convexly independent* if, for every subset $J \\subseteq I$ and every $i \\in I$,\n$$x_i \\in \\operatorname{conv}\\set{x_j : j \\in J} \\implies i \\in J,$$\nwhere $\\operatorname{conv}$ denotes the [[convex hull|0FQSDC2EQ2]].\nA subset $S \\subseteq V$ is *in convex position* if the family $(x)_{x \\in S}$ is convexly independent, that is, if no $x \\in S$ lies in $\\operatorname{conv}(S \\setminus \\set{x})$.\n\n## Remarks\n\nExample: the points $(0,0)$, $(1,0)$, $(0,1)$, $(1,1)$ of $\\R^2$ are in convex position.\nEach of them is a vertex of the unit square, and no vertex is a convex combination of the other three.\nFor instance, the other three points satisfy $x + y \\le 1$, so all their convex combinations do, while $(1,1)$ does not.\nThe other vertices are handled by symmetry.\n\nNon-example: add the point $(\\frac12, \\frac12)$.\nIt equals $\\frac12 (1,0) + \\frac12 (0,1)$, so it lies in the convex hull of the other points.\n\nA convexly independent family is injective, because $x_i = x_j$ with $i \\neq j$ puts $x_i$ in the hull of $\\set{x_j}$.\nA finite set $S$ is in convex position if and only if every point of $S$ is an [[extreme point|3MSGFJM2NX]] of $\\operatorname{conv}(S)$, that is, a vertex of the polytope $\\operatorname{conv}(S)$.\nAffinely independent families are convexly independent, but not conversely: the four corners of a square are affinely dependent.\n",
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