Cutting a convex body into fat parts and approximating Euclidean distance by graph distances

Can one construct a graph $G$ on the set of integer points ${\mathbb{Z}}^2$ in the plane such that the length of the shortest path between any two vertices of $G$ differs from their Euclidean distance by at most an absolute constant? This question of Benjamini, Erd\H os, Kleiner, Kozma, Schramm, and the first-named author has been open for a long time. We give an affirmative answer to a weaker form of this question, based on the following geometric statement, which is of independent interest. There exists a constant $c>0$ such that for every $i=1,2,\dots ,$ every $\rho$-fat plane convex set $S$ can be cut into $2^i$ convex pieces of equal area, each of which is at least $c\rho$-fat. (A convex set is $\rho$-fat if the ratio of its inradius to its circumradius is at least $\rho$.) We prove that there exists an (unweighted) spanning subgraph $G$ of an enlarged copy of ${\mathbb{Z}}^2$ such that, for every pair of vertices at Euclidean distance $d$, their shortest-path distance in $G$ lies between $d-O(1)$ and $d+o(d^{5\mathrm{/}6})$. The same bound can be achieved by a planar graph with vertex set ${\mathbb{Z}}^2$, in which every edge joins two vertices at Euclidean distance at most 2.