Given two intervals $ X = [x_{min}, x_{max}], Y = [y_{min}, y_{max}]$ and an function $f: X\times Y\rightarrow \{0,1\}$ which confirms, whether a point is in or out of an unknown $k$-sided geometric shape $s$ in polynomial-time. Furthermore, one point $p$ known to be in the shape, $k$ is given, and all of $s$ lies within the rectangle induced by the intervals.
Is there a polynomial-time (in $k$) algorithm for computing the convex hull of said shape or an approximation algorithm which outputs a shape which covers as much of the surface of $s$ while not covering any surface outside of $s$?
