Local Differences Determined by Convex Sets

This paper introduces a new problem concerning additive properties of convex sets. Let $S= \{s_1 < \dots <s_n \}$ be a set of real numbers and let $D_i(S)= \{s_x-s_y: 1 \leq x-y \leq i\}$. We expect that $D_i(S)$ is large, with respect to the size of $S$ and the parameter $i$, for any convex set $S$. 
We give a construction to show that $D_3(S)$ can be as small as $n+2$, and show that this is the smallest possible size. On the other hand, we use an elementary argument to prove a non-trivial lower bound for $D_4(S)$, namely $|D_4(S)| \geq \frac{5}{4}n -1$. For sufficiently large values of $i$, we are able to prove a non-trivial bound that grows with $i$ using incidence geometry.