= McShane extension theorem
{c}
{title2=$F(x)=\inf_{a\in A}\{f(a)+Ld(x,a)\}$}
Let $A$ be a nonempty <subset> of a <metric space> $X$, and let $f:A\to\mathbb R$ satisfy a <Lipschitz bound> $L$. Then
$$
F(x)=\inf_{a\in A}\{f(a)+Ld(x,a)\},\qquad
G(x)=\sup_{a\in A}\{f(a)-Ld(x,a)\}
$$
are real-valued <Lipschitz functions> on all of $X$, agree with $f$ on $A$, and obey the same bound $L$. Every other extension $H$ satisfying that bound lies between them, $G\leq H\leq F$. Despite the <infimum> in its formula, $F$ is the largest such extension: every given sample supplies an upper constraint $H(x)\leq f(a)+Ld(x,a)$.
Take three real samples $f(0)=0$, $f(1)=1/2$, $f(2)=0$, and allow the bound $L=1$. Their least <Lipschitz constant> is only $1/2$, so this is deliberately a looser permitted bound. The two extensions are
$$
F(x)=\min\{|x|,\tfrac12+|x-1|,|x-2|\},\qquad
G(x)=\max\{-|x|,\tfrac12-|x-1|,-|x-2|\}.
$$
At $x=1/2$, the allowable <interval> is $0=G(1/2)\leq H(1/2)\leq F(1/2)=1/2$. The <interpolant> obtained by <linear interpolation> is a <piecewise linear function> and has value $1/4$ there. Extending it constantly outside $[0,2]$ obeys the same bound on the entire <real line>. At $x=5/2$, the allowable <interval> is $[-1/2,1/2]$. Thus the data and the bound control possible values without selecting a unique curve.
\Image[/lipschitz-extension-envelopes.png]
{title=Each sample constrains a <Lipschitz function>; the upper and lower envelopes bound every allowed extension}
The construction is useful when data are known only on a <subset>: it extends them without increasing an allowed <Lipschitz bound>. A short justification explains why the envelopes work. Fix $a_0\in A$. The <triangle inequality> and the bound on $f$ imply
$$
f(a_0)-Ld(x,a_0)\leq f(a)+Ld(x,a)
\qquad(a\in A),
$$
so the <infimum> is finite below; the term $a=a_0$ makes it finite above. At a sample $x=a_0$, all upper terms are at least $f(a_0)$, while that term gives equality. Finally, $d(x,a)\leq d(y,a)+d(x,y)$ gives $F(x)\leq F(y)+Ld(x,y)$ after taking an <infimum>; exchanging $x,y$ proves the bound. The lower envelope follows by applying the same argument to $-f$.
E. J. McShane's paper \i[Extension of range of functions] appeared in the \i[Bulletin of the American Mathematical Society], volume 40, in 1934, pp. 837–842. https://www.ams.org/books/surv/012/surv012-endmatter.pdf[AMS bibliography: E. J. McShane, Extension of range of functions]{ref} https://doi.org/10.1090/S0002-9904-1934-05978-0[McShane's original publication]{ref}
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