Addenda and erratum to "Mapping radii of metric spaces" by George M. Bergman

Addenda:

While correcting the galley proofs, the question occurred to me:  For  X  a metric space and  n > 1,  is there in general any normed vector space  V  of dimension  n  making  map-rad_V(X)  as small as  diam(X) / 2  (its mapping radius in a 1-dimensional space)?  An affirmative answer follows from the corollary to the following easily verified result: 

Lemma 34.  If  Y₁, ... , Y_n  and  X  are metric spaces, and we give the product space  Y = Y₁× ... ×Y_n  the  sup  norm, then  map-rad_Y (X) = sup_i map-rad _Yi (X). \qed

Corollary 35.  If  V  denotes  Rⁿ  with the sup norm, and  X  is any metric space, then  map-rad_V (X) = diam(X) / 2 . \qed

Corresponding results hold for infinite products, but require a little more care to state, because the "sup norm" on an unrestricted infinite product can take on the value infinity; so one needs to, e.g., take a basepoint in each space, and consider the subspace of the product consisting of elements whose coordinates have bounded distance from these basepoints.  In  R^N  this is the space of bounded real-valued sequences  l^∞.  For general metric spaces, I don't know whether it has a name.

On a different subject:  the operator  D  of section 6, taking measures of finite support on  X × X  to measures of finite support on  X,  has the look of a boundary operator.  Weaver and I discussed this, but did not know what to make of it.  Clearly, we are not using it in the way boundary operators are usually used:  we are not interested in its kernel or cokernel.  Does it belong to a wider system of operators taking measures on  Xⁿ  to measures on  Xⁿ-1,  and if so, how might one use these? 

Finally, both John Lott and Jonathan Dahl have suggested that in some or all of the results I get using convex linear combinations of points in normed vector spaces, one may be able to generalize these vector spaces to Hadamard spaces.  In these, one has a concept of the midpoint of a pair of points.  Iterated midpoints give operators behaving like averages of  2ⁿ  points, and convex linear combinations with arbitrary real coefficients can be mimicked by limits, as  n  approaches infinity, of such iterated averages. 

Erratum:  In the abstract, the words "normed metric spaces" should be "normed vector spaces". 

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