Lipschitz extension theorems with explicit constants

In this mostly expository article, we give streamlined proofs of several well-known Lipschitz extension theorems. We pay special attention to obtaining statements with explicit expressions for the extension constants. One of our main results is an explicit version of a very general Lipschitz extension theorem of Lang and Schlichenmaier. A special case of the theorem reads as follows. We prove that if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>X</m:mi> </m:math> X is a metric space and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>A</m:mi> <m:mo>⊂</m:mo> <m:mi>X</m:mi> </m:math> A⊂ X satisfies the condition <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mstyle> <m:mspace width="0.1em"/> <m:mtext>Nagata</m:mtext> <m:mspace width="0.1em"/> </m:mstyle> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi>c</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {0.1em}Nagata{0.1em}(n,c) , then any 1-Lipschitz map <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>f</m:mi> <m:mo>:</m:mo> <m:mi>A</m:mi> <m:mo>→</m:mo> <m:mi>Y</m:mi> </m:math> f:A→ Y to a Banach space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>Y</m:mi> </m:math> Y admits a Lipschitz extension <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>F</m:mi> <m:mo>:</m:mo> <m:mi>X</m:mi> <m:mo>→</m:mo> <m:mi>Y</m:mi> </m:math> F:X→ Y whose Lipschitz constant is at most <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mstyle> <m:mspace width="0.1em"/> <m:mtext>1,000</m:mtext> <m:mspace width="0.1em"/> </m:mstyle> <m:mo>⋅</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>c</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>⋅</m:mo> <m:msub> <m:mrow> <m:mi>log</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {0.1em}1,000{0.1em}· (c+1)· {log }₂(n+2) . By specifying to doubling metric spaces, this recovers an extension result of Lee and Naor. We also revisit another theorem of Lee and Naor by showing that if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>A</m:mi> <m:mo>⊂</m:mo> <m:mi>X</m:mi> </m:math> A⊂ X consists of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> </m:math> n points, then Lipschitz extensions as above exist with a Lipschitz constant of at most <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>600</m:mn> <m:mo>⋅</m:mo> <m:mi>log</m:mi> <m:mi>n</m:mi> <m:mo>⋅</m:mo> <m:msup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>log</m:mi> <m:mi>log</m:mi> <m:mi>n</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> 600· log n· {(log log n)}⁻¹ .