Explicit construction of a convex metric

Question

Let $(X,d)$ be a compact, connected, locally connected, locally compact metric space.

A result of Bing and Moise (independently) states that $(X,d)$ admits a topology preserving convex metric i.e., a metric $d'$ such that if $p,q \in X$, there exists a point $r\in X$ such that $d'(p,r) = d'(q,r) = d'(p,q)/2$.

I would like to understand the construction of the convex metric $d'$.

The main point that I can gather is that, the authors define a sequence $E_1(p,q),\, E_2(p,q),\dots$ of functions on pairs of points of $X$ such that $E(p,q) = \lim E_i(p,q)$ is a convex metric on $X$.

However, I do not understand how to explicitly compute these $E_i(p,q)$, as the proofs of Bing and Moise are rather complicated and involved.

Question. Can someone explain how to explicitly compute these $E_i(p,q)$? Or provide a less technical description of these $E_i(p,q)$? Or provide a reference to where this convex metric is unraveled a bit?

Thanks!

References

Bing, Partitioning a Set, Bull. Amer. Math Soc., Vol 55 (1949) pp. 1101-1110

Moise, Grille decomposition and convexification theorems for compact locally connected continua, Bull. Amer. Math Soc., Vol 55 (1949) pp. 1111-1121

Answer 1

As far as I know the most accessible proof of Bing's construction is in Section 2 of the paper:

J. C. Mayer, L. G. Oversteegen, E, D. Tymchatyn, The Menger curve. Characterization and extension of homeomorphisms of non-locally-separating closed subsets. Dissertationes Math. (Rozprawy Mat.) 252 (1986), 45 pp.

The paper can be downloaded from:

http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1d60f6d5-8102-4734-9186-d4b4bfd8f2dc

The paper does not address construction of a convex metric, but many years ago I wanted to learn the proof of the existence of a convex metric (I never did) and this is what Tymchatyn wrote to me (in 2007):

Once you have appropriate partitions the construction of a convex metric is relatively easy. Bing's 1952 BAMS paper Partitioning Continuous Curves is the place to read it.

There is a proof of the partitioning theorem for Peano continua in the book of Hall and Spencer,Elementary Topology,Wiley,1955. As I recall the proof there is the one given by Bing. The first part of Bing's proof can be simplified a bit using a Peano map of the interval onto a Peano continuum $X$ to get a decomposition of $X$ into finitely many nice bricks between a closed subset $A$ of $X$ and a point $p$ outside $A$.

Bing assigns weights to elements in a defining sequence of partitions $U_i$ of $X$ in such a way that shortest chains in the partitions between two points converge to a distance which gives the convex metric.