  Very recently, Galashin, Postnikov, and Williams introduced the notion of
higher secondary polytopes, generalizing the secondary polytope of Gelfand,
Kapranov, and Zelevinsky. Given an $n$-point configuration $\mathcal{A}$ in
$\mathbb{R}^{d-1}$, they define a family of convex $(n-d)$-dimensional
polytopes $\widehat{\Sigma}_{1}, \ldots, \widehat{\Sigma}_{n-d}$. The
$1$-skeletons of this family of polytopes are the flip graphs of certain
combinatorial configurations which generalize triangulations of $\text{conv}
\mathcal{A}$. We restrict our attention to $d=2$. First, we relate the
$1$-skeleton of the Minkowski sum $\widehat{\Sigma}_{k} +
\widehat{\Sigma}_{k-1}$ to the flip graph of "hypertriangulations" of the
deleted $k$-sum of $\mathcal{A}$ when $\mathcal{A}$ consists of distinct
points. Second, we compute the diameter of $\widehat{\Sigma}_{k}$ and
$\widehat{\Sigma}_{k}+\widehat{\Sigma}_{k-1}$ for all $k$.
