  We study the two-dimensional geometric knapsack problem for convex polygons.
Given a set of weighted convex polygons and a square knapsack, the goal is to
select the most profitable subset of the given polygons that fits
non-overlappingly into the knapsack. We allow to rotate the polygons by
arbitrary angles. We present a quasi-polynomial time $O(1)$-approximation
algorithm for the general case and a polynomial time $O(1)$-approximation
algorithm if all input polygons are triangles, both assuming polynomially
bounded integral input data. Also, we give a quasi-polynomial time algorithm
that computes a solution of optimal weight under resource augmentation, i.e.,
we allow to increase the size of the knapsack by a factor of $1+\delta$ for
some $\delta>0$ but compare ourselves with the optimal solution for the
original knapsack. To the best of our knowledge, these are the first results
for two-dimensional geometric knapsack in which the input objects are more
general than axis-parallel rectangles or circles and in which the input
polygons can be rotated by arbitrary angles.
