  In this paper we prove that any convex body of the d-dimensional Euclidean
space (d>1) possesses at least one shortest generalized billiard trajectory
moreover, any of its shortest generalized billiard trajectories is of period at
most d+1. Actually, in the Euclidean plane we improve this theorem as follows.
A disk-polygon with parameter r>0 is simply the intersection of finitely many
(closed) circular disks of radii r, called generating disks, having some
interior point in common in the Euclidean plane. Also, we say that a
disk-polygon with parameter r>0 is a fat disk-polygon if the pairwise distances
between the centers of its generating disks are at most r. We prove that any of
the shortest generalized billiard trajectories of an arbitrary fat disk-polygon
is a 2-periodic one. Also, we give a proof of the analogue result for
{\epsilon}-rounded disk-polygons obtained from fat disk-polygons by rounding
them off using circular disks of radii {\epsilon}>0. Our theorems give partial
answers to the very recent question raised by S. Zelditch on characterizing
convex bodies whose shortest periodic billiard trajectories are of period 2.
