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  "body": "\n## Definition\n\nA set $C \\subseteq \\mathbb{R}^n$ is convex if $t x + (1 - t) y \\in C$ for all $x, y \\in C$ and all $t \\in [0, 1]$.\nLet $C \\subseteq \\mathbb{R}^n$ be convex.\nA function $f \\colon C \\to \\mathbb{R}$ is convex if, for all $x, y \\in C$ and all $t \\in [0, 1]$,\n$$f(t x + (1 - t) y) \\le t f(x) + (1 - t) f(y).$$\nThe function $f$ is strictly convex if the inequality is strict for all $x \\ne y$ and all $t \\in (0, 1)$.\n\n## Remarks\n\nIf $C$ is open and $f$ is differentiable on $C$, then $f$ is convex if and only if $f(y) \\ge f(x) + \\nabla f(x) \\cdot (y - x)$ for all $x, y \\in C$.\nLet $C$ be open and let $f$ be of class $C^2$. Then $f$ is convex if and only if its [[Hessian matrix|0SCSQWXRA4]] is positive semidefinite at each point of $C$.\nEach local minimum of a convex function is a global minimum, and a strictly convex function has at most one global minimum.\nA function can be convex in each variable separately but not convex. The function $f(x, y) = x y$ is linear in each variable, but $f(t, -t) = -t^2$ is not convex.\n\n## Explanations\n\n### L1 human\n\nA convex function is bowl-shaped: the straight segment between two points of its graph never goes below the graph.\nExamples are $\\|x\\|^2$ and, on all of $\\mathbb{R}^n$, each linear function.\n",
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