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](https://projecteuclid.org/euclid.bams/1183514375), *Partitioning a Set*, Bull. Amer. Math Soc., Vol 55 (1949) pp. 1101-1110


[Moise](https://projecteuclid.org/euclid.bams/1183514376), *Grille decomposition and convexification theorems for compact locally connected continua*, Bull. Amer. Math Soc., Vol 55 (1949) pp. 1111-1121



Explicit construction of a convex metric
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.*


