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

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