Geometric inequality by convexity

Question

Let $A$, $B$ and $C$ be the vertices of a triangle and let $D$ be a point in its interior. We denote the lenghts of the line segments $AD$, $BD$ and $CD$ by $x$, $y$, $z$ respectively. It is to be shown that the sum $x+y+z$ is less than the triangle´s perimeter. I know this questions has been asked before and there are multiple elementary solutions, but I have been trying to prove this using convexity, the maximum having to be assumed on a vertex, but without success. I´d appreciate solutions.