Lipschitz Extension Constants Equal Projection Constants

For a Banach space V we define its Lipschitz extension constant, LE(V ), to be the infimum of the constants c such that for every metric space (Z, ρ), every X ⊂ Z, and every f : X → V , there is an extension, g, of f to Z such that L(g) ≤ cL(f), where L denotes the Lipschitz constant. The basic theorem is that when V is finite-dimensional we have LE(V ) = PC(V ) where PC(V ) is the well-known projection constant of V . We obtain some direct consequences of this theorem, especially when V = Mn(C). We then apply known techniques for calculating projection constants, involving averaging of projections, to calculate LE((Mn(C))sa). We also discuss what happens if we also require that ‖g‖∞ = ‖f‖∞. In my exploration of the relationship between vector bundles and Gromov– Hausdorff distance [20] I need to be able to extend matrix-valued functions from a closed subset of a compact metric space to the whole metric space, with as little increase of the Lipschitz constant as possible. There is a substantial literature concerned with extending Lipschitz functions, but I have had difficulty finding there the facts which I need. The purpose of this largely expository paper is to describe and employ a very strong relationship between the Lipschitz extension problem and what is referred to as the “projection constant” for finite-dimensional Banach spaces. This permits us to bring to bear on the Lipschitz extension problem the quite substantial literature concerning projection constants. This then provides the facts which I need, as well as other interesting facts. In Section 1 we introduce what we call the Lipschitz extension constant, LE(V ), of a Banach space V . I have not found exactly this definition in the literature, although there are definitions very close to it. We also recall the well-known definition of the projection constant, PC(V ), of a Banach space V . The basic theorem is that if V is finite-dimensional, then LE(V ) = PC(V ). I have not found this theorem stated in the literature, probably because LE(V ) is not defined in the literature, but I am told that this theorem is well-known to specialists on the geometry of Banach spaces. In Section 2 we give the proof that LE(V ) ≤ PC(V ), while in Section 3 we give the proof that PC(V ) ≤ LE(V ), thus proving the basic theorem. In Section 4 we give some consequences of the basic theorem which come directly from using facts about projection constants that are available in the literature. One of these consequences is that LE(C) = 4/π. Another of these consequences is the 2000 Mathematics Subject Classification. 46B20; 26A16.