  A greedy embedding of a graph $G = (V,E)$ into a metric space $(X,d)$ is a
function $x : V(G) \to X$ such that in the embedding for every pair of
non-adjacent vertices $x(s), x(t)$ there exists another vertex $x(u)$ adjacent
to $x(s)$ which is closer to $x(t)$ than $x(s)$. This notion of greedy
embedding was defined by Papadimitriou and Ratajczak (Theor. Comput. Sci.
2005), where authors conjectured that every 3-connected planar graph has a
greedy embedding (possibly planar and convex) in the Euclidean plane. Recently,
greedy embedding conjecture has been proved by Leighton and Moitra (FOCS 2008).
However, their algorithm do not result in a drawing that is planar and convex
for all 3-connected planar graph in the Euclidean plane. In this work we
consider the planar convex greedy embedding conjecture and make some progress.
We derive a new characterization of planar convex greedy embedding that given a
3-connected planar graph $G = (V,E)$, an embedding $x: V \to \bbbr^2$ of $G$ is
a planar convex greedy embedding if and only if, in the embedding $x$, weight
of the maximum weight spanning tree ($T$) and weight of the minimum weight
spanning tree ($\func{MST}$) satisfies $\WT(T)/\WT(\func{MST}) \leq
(\card{V}-1)^{1 - \delta}$, for some $0 < \delta \leq 1$.
