  We prove Gagliardo-Nirenberg inequalities with three weights -- verifying a
joint concavity condition -- on open convex cones of $\mathbb R^n$. If the
weights are equal to each other the inequalities become sharp and we compute
explicitly the sharp constants. For a certain range of parameters we can
characterize the class of extremal functions; in this case, we also show that
the sharpness in the main three-weighted Gagliardo-Nirenberg inequality implies
that the weights must be equal up to some constant multiplicative factors. Our
approach uses optimal mass transport theory and a careful analysis of the joint
concavity condition of the weights. As applications we establish sharp weighted
$p$-log-Sobolev, Faber-Krahn and isoperimetric inequalities with explicit sharp
constants.
