  This paper is devoted to the examination of the properties of the string
construction for the Birkhoff billiard. Based on purely geometric
considerations, string construction is suited to provide a table for the
Birkhoff billiard, having the prescribed caustic. Exploiting this framework
together with the properties of convex caustics, we give a geometric proof of a
result by Innami first proved in 2002 by means of Aubry-Mather theory. In the
second part of the paper we show that applying the string construction one can
find a new collection of examples of $C^2$-smooth convex billiard tables with a
non-smooth convex caustic.
