least 1/2 and have edge density $e$. From the convexity of $I_0$, we know that the graphon minimizing $I$ must be zero on a set of measure 1/2 and must be constant on the rest. Thus $g_0$ is a minimizer, and $\min_{F_0}I(g) = I(g_0) = 
I_0(2e)/2$.

Now suppose that $g$ is another minimizer. Since $g$ is zero on a set of measure 1/2 and is $2e$ on a set of measure 1/2, $\bar g$ is 1 on a set of measure 1/2, and so describes a graphon with edge density 1/2 and no triangles. This means that $\bar g$ describes a complete bipartite graph with the two parts having the same measure. That is, $\bar g$ is equivalent to the graphon that equals 1 if $x<\frac{1}{2}<y$ or $y< \frac{1}{2} < x$ and is zero everywhere else. But then $g=2e\bar g$ is equivalent to $g_0$.

The situation on the curved scallops is slightly more complicated. Pick an integer $\ell>1$. (The case $\ell=1$ just gives us our first scallop.) If $e
\in \left[1-\frac{1}{\ell}, 1-\frac{1}{\ell+1}\right ]$, then any graph $G$ with edge density $e$ and the minimum number of triangles has to take the following form (see [@PR] for the history). Let $$c 