TITLE: Shadow boundary on convex body in $\mathbb{R}^3$
QUESTION [7 upvotes]: Let $S$ be the surface of a compact, convex, smooth ($C^\infty$) body in $\mathbb{R}^3$,
with strictly positive Gaussian curvature at every point of $S$.
Fix a direction $z$ in a Cartesian coordinate system,
and consider all the lines parallel to $z$ and tangent to $S$,
which form a topological cylinder enclosing $S$,
touching $S$ on the shadow boundary resulting from a light source at $z=+\infty$
(yellow in the figure below).
Parametrize these lines from $s=0$ to $s=1$ around the cylinder,
and let $h(s)$ be the height of the point of tangency
to $S$ above the $xy$-plane, orthogonal to $z$.
My question is:

Can $h(s)$ have an arbitrarily large number of local maxima and minima?


I am interested to learn if this shadow-boundary curve is "well-behaved" in some sense,
for smooth convex bodies.
Thanks for pointers/suggestions/counterexmaples!

REPLY [2 votes]: This is not an answer but a shadow  tangent  Mohammad Ghomi has a wonderful paper
concerning a converse question:  what are neccessary and sufficient conditions
on the shadows which insure   the surface to be convex.  His answer:
for all projection directions, the shadow must be connected and simply connected.
See: 
Solution to the shadow problem in 3-space, in Minimal Surfaces, Geometric Analysis and Symplectic Geometry, Adv. Stud. Pure Math, 34 (2002) 129-142. Which you can find on his web page.