We consider the problem of enumerating all minimal transversals of a hypergraph. In general, it is not known whether one can enumerate all (inclusion-wise) minimal transversals in output polynomial time. We show that if the hypergraph has a "balanced subdivision", then we can indeed enumerate its minimal transversals in time polynomial in the output size. We finally show that several geometrically induced hypergraphs do have such a balanced subdivision. This gives polynomial time algorithms for them.