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

Pacific Journal of Mathematics, 236 (2008) 223-261.


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-radV(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, ... , Yn  and  X  are metric spaces, and we give the product space  Y = Y1× ... ×Yn  the  sup  norm, then  map-radY (X) = supi map-rad Yi (X). \qed

Corollary 35.  If  V  denotes  Rn  with the sup norm, and  X  is any metric space, then  map-radV (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  RN  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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