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

Pacific Journal of Mathematics, **236** (2008) 223-261.
**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
Y*_{1}, ... , *Y*_{n} and
X are metric spaces, and we give the product space Y
= Y_{1}× ... ×*Y*_{n}
the sup *norm, then*
map-rad_{Y} (*X*) = sup_{i} map-rad_{ Yi} (*X*). \qed

**Corollary 35.** *If V
denotes ***R**^{n} 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 *X*^{n} to measures on
*X*^{n}-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 2^{n} 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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