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  "body": "\n## Summary\n\nThe proximal operator of a convex function $f$ on a Hilbert space sends a point $v$ to a nearby point where $f$ is small.\nThe new point minimizes $f(x) + \\tfrac12 \\norm{x - v}^2$, which balances the two aims.\nIt generalizes the projection onto a closed convex set, which is the proximal operator of the indicator function of the set.\nProximal operators are firmly nonexpansive, and they are the building blocks of proximal gradient methods and other splitting algorithms in convex optimization.\n\n## Explanations\n\n### L1 human\n\nImagine standing at a point $v$ and wanting to move to a place where $f$ is small, but each step away costs you.\nThe proximal point is the best compromise: it minimizes $f(x)$ plus half the squared distance from $x$ to $v$.\nIf $f$ is $0$ on a convex region and $+\\infty$ outside, the compromise is simply the nearest point of the region.\nFor $f(x) = \\abs{x}$ on $\\R$, the proximal operator moves $v$ toward $0$ by $1$ and stops at $0$: it sends $3$ to $2$ and $0.4$ to $0$.\n",
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