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  "body": "\n## Definition\n\nLet $V$ be a real [[vector space|000AYPA9AJ]] and let $A, B \\subseteq V$.\nThe *convex join* of $A$ and $B$ is the set\n$$\\operatorname{join}(A, B) = \\bigcup_{a \\in A,\\ b \\in B} [a, b],$$\nwhere $[a, b]$ is the [[line segment|VQ4Q2V45RP]] from $a$ to $b$.\n\n## Remarks\n\nExample: in $\\R^2$, let $A = \\set{(0, 0)}$ and $B = [(1, 0), (1, 1)]$.\nThe join is the union of the segments from the origin to the points $(1, s)$ with $0 \\le s \\le 1$, which is the closed triangle with vertices $(0,0)$, $(1,0)$ and $(1,1)$.\n\nIf $A$ and $B$ are nonempty [[convex sets|02AS45T3H9]], then $\\operatorname{join}(A, B)$ equals the [[convex hull|0FQSDC2EQ2]] of $A \\cup B$.\nIndeed, a convex combination of points of $A \\cup B$ can be regrouped as $(1 - t) a + t b$ with $a \\in A$ and $b \\in B$.\n\nNon-example of that identity: if $A = \\emptyset$ and $B = \\set{b}$, then $\\operatorname{join}(A, B) = \\emptyset$, while the convex hull of $A \\cup B$ is $\\set{b}$.\nSo the nonemptiness hypothesis cannot be dropped.\n",
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