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  "body": "\n## Definition\n\nLet $X$ be a [[normed vector space|03J5RAKB30]] and let $K \\subseteq X$ be nonempty.\nThe set $K$ is a *Chebyshev set* if every $x \\in X$ has exactly one [[best approximation|7N4988HN3P]] from $K$, that is, exactly one $k \\in K$ with $\\norm{x - k} = \\inf_{y \\in K} \\norm{x - y}$.\nThe map $P_K \\colon X \\to K$ sending $x$ to this point is the *metric projection* onto $K$.\n\n## Remarks\n\nExample: in $\\R^2$ with the Euclidean norm, the closed unit disc $D$ is a Chebyshev set.\nFor $\\norm{x} \\le 1$ the nearest point is $x$ itself.\nFor $\\norm{x} \u003e 1$ it is $x / \\norm{x}$ and no other point of $D$, since the triangle inequality is strict off the ray through $x$.\n\nNon-example: the unit circle in $\\R^2$ is not a Chebyshev set, because the origin has every point of the circle as a nearest point.\n\nA Chebyshev set is closed, since a point of the closure not in $K$ would have distance $0$ but no nearest point.\nEvery nonempty closed convex set in a Hilbert space is a Chebyshev set, by the [[Hilbert projection theorem|J5YNTHQMFX]].\nIn $\\R^n$ with the Euclidean norm the converse holds: every Chebyshev set is convex.\nWhether every Chebyshev set in an infinite-dimensional Hilbert space is convex is an open problem.\n",
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