  Let $\mathcal{B}$ be a nonempty homothecy invariant collection of convex sets
of positive finite measure in $\mathbb{R}^2$. Let $M_\mathcal{B}$ be the
geometric maximal operator defined by $$M_\mathcal{B}f(x) = \sup_{x \in R \in
\mathcal{B}}\frac{1}{|R|}\int_R |f|\;.$$ We show that either $M_\mathcal{B}$ is
bounded on $L^p(\mathbb{R}^2)$ for every $1 < p \leq \infty$ or that
$M_\mathcal{B}$ is unbounded on $L^p(\mathbb{R}^2)$ for every $1 \leq p <
\infty$. As a corollary, we have that any density basis that is a homothecy
invariant collection of convex sets in $\mathbb{R}^2$ must differentiate
$L^p(\mathbb{R}^2)$ for every $1 < p \leq \infty$.
