  For each odd $n \geq 3$, we construct a closed convex hypersurface of
$\mathbb{R}^{n+1}$ that contains a non-degenerate closed geodesic with Morse
index zero. A classical theorem of J. L. Synge would forbid such constructions
for even $n$, so in a sense we prove that Synge's theorem is "sharp." We also
construct stable figure-eights: that is, for each $n \geq 3$ we embed the
figure-eight graph in a closed convex hypersurface of $\mathbb{R}^{n+1}$, such
that sufficiently small variations of the embedding either preserve its image
or must increase its length. These index-zero geodesics and stable
figure-eights are mainly derived by constructing explicit billiard trajectories
with "controlled parallel transport" in convex polytopes.
