  In this note we study the distribution of real inflection points among the
ovals of a real non-singular hyperbolic curve of even degree. Using Hilbert's
method we show that for any integers $d$ and $r$ such that $4\leq r \leq
2d^2-2d$, there is a non-singular hyperbolic curve of degree $2d$ in $\mathbb
R^2$ with exactly $r$ line segments in the boundary of its convex hull. We also
give a complete classification of possible distributions of inflection points
among the ovals of a maximally inflected non-singular hyperbolic curve of
degree $6$.
