Lipschitz extension constants equal projection constants

Abstract

For a Banach space $V$ we define its Lipschitz extension constant, $\cL\cE(V)$, to be the infimum of the constants $c$ such that for every metric space $(Z,\rho )$, every $X\subset Z$, and every $f:X\to V$, there is an extension, $g$, of $f$ to $Z$ such that $L(g)\le cL(f)$, where $L$ denotes the Lipschitz constant. The basic theorem is that when $V$ is finite-dimensional we have $\cL\cE(V) = \cP\cC(V)$ where $\cP\cC(V)$ is the well-known projection constant of $V$. We obtain some direct consequences of this theorem, especially when $V = M_n(\bC)$. We then apply techniques for calculating projection constants, involving averaging projections, to calculate $\cL\cE((M_n(\bC))^{sa})$. We also discuss what happens if we also require that $\Vert g{\Vert }_{\infty }=\Vert f{\Vert }_{\infty }$.

16 pages. Three very minor mathematical typos corrected. Intended for the proceedings of GPOTS05