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 Incoming links: McShane extension theorem

f : A \to R

L

when

∣ f (y) - f (z) ∣ \leq L d (y, z)

for all

y, z \in A

Fix

y_{0} \in A

. The Lipschitz inequality and the triangle inequality imply

f (y) + L d (x, y) \geq f (y_{0}) - L d (y, y_{0}) + L d (x, y) \geq f (y_{0}) - L d (x, y_{0}),

while choosing

y = y_{0}

gives a finite upper bound. Hence the infimum defining

F (x)

x \in A

, then

f (x) \leq f (y) + L d (x, y)

for every

y \in A

, while

y = x

gives the reverse inequality after taking the infimum. Thus

F (x) = f (x)

. For arbitrary

x, z \in X

f (y) + L d (x, y) \leq f (y) + L d (z, y) + L d (x, z) .

Taking infima gives

F (x) \leq F (z) + L d (x, z)

; interchanging

x, z

gives

∣ F (x) - F (z) ∣ \leq L d (x, z) .

This is the McShane extension theorem: $F$ extends $f$ without increasing its Lipschitz constant.