  We study the {\sc Clique} problem in classes of intersection graphs of convex
sets in the plane. The problem is known to be NP-complete in convex-set
intersection graphs and straight-line-segment intersection graphs, but solvable
in polynomial time in intersection graphs of homothetic triangles. We extend
the latter result by showing that for every convex polygon $P$ with sides
parallel to $k$ directions, every $n$-vertex graph which is an intersection
graph of homothetic copies of $P$ contains at most $n^{k}$ inclusion-wise
maximal cliques. We actually prove this result for a more general class of
graphs, the so called $k_{\text{DIR}}-\text{CONV}$, which are intersection
graphs of convex polygons whose sides are parallel to some fixed $k$
directions. Moreover, we provide some lower bounds on the numbers of maximal
cliques, discuss the complexity of recognizing these classes of graphs and
present a relationship with other classes of convex-set intersection graphs.
Finally, we generalize the upper bound on the number of maximal cliques to
intersection graphs of higher-dimensional convex polytopes in Euclidean space.
