Just empirically, I believe the OP's $e^{-x}$ example has the property that ray reflections become increasingly vertical, and so will not reach arbitrarily large $x$:
![exconvex][1]
(I did not, however, perform the calculations to prove this.)
![exconvex][1]
(I did not, however, perform the calculations to prove this.)
Joseph O'Rourke
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