Antimedian graphs are introduced as the graphs in which for every triple
of vertices there exists a unique vertex x that maximizes the sum of the
distances from x to the vertices of the triple. The Cartesian product of
graphs is antimedian if and only if its factors are antimedian. It is proved
that multiplying a non-antimedian vertex in an antimedian graph yields
a larger antimedian graph. Thin even belts are introduced and proved to
be antimedian. A characterization of antimedian trees is given that leads
to a linear recognition algorithm.