Escape Orbits for Non-Compact Flat Billiards

It is proven that, under some conditions on $f$, the non-compact flat billiard $\mathrm{\Omega }=\{(x,y)\in {\mathbb{R}}_0^{+}\times {\mathbb{R}}_0^{+};0\le y\le f(x)\}$ has no orbits going {\em directly} to $+\mathrm{\infty }$. The relevance of such sufficient conditions is discussed.