  A conjecture by Rafla from 1988 asserts that every simple drawing of the
complete graph $K_n$ admits a plane Hamiltonian cycle. It turned out that
already the existence of much simpler non-crossing substructures in such
drawings is hard to prove. Recent progress was made by Aichholzer et al. and by
Suk and Zeng who proved the existence of a plane path of length $\Omega(\log n
/ \log \log n)$ and of a plane matching of size $\Omega(n^{1/2})$ in every
simple drawing of $K_{n}$.
  Instead of studying simpler substructures, we prove Rafla's conjecture for
the subclass of convex drawings, the most general class in the convexity
hierarchy introduced by Arroyo et al. Moreover, we show that every convex
drawing of $K_n$ contains a plane Hamiltonian path between each pair of
vertices (Hamiltonian connectivity) and a plane $k$-cycle for each $3 \leq k
\leq n$ (pancyclicity), and present further results on maximal plane
subdrawings.
