  Rational Diophantine triples, i.e. rationals a,b,c with the property that
ab+1, ac+1, bc+1 are perfect squares, are often used in construction of
elliptic curves with high rank. In this paper, we consider the opposite problem
and ask how small can be the rank of elliptic curves induced by rational
Diophantine triples. It is easy to find rational Diophantine triples with
elements with mixed signs which induce elliptic curves with rank 0. However,
the problem of finding such examples of rational Diophantine triples with
positive elements is much more challenging, and we will provide the first such
known example.
