Electric field due to a charged irregularly shaped balloon

There is a question in my textbook that says:

A rubber balloon is given a charge $Q$ distributed uniformly over its surface. Is the field inside the balloon zero everywhere if the balloon does not have a spherical surface?

Intuitively i think the answer is no, but i am not able to prove it conclusively. The best i could say is that if the balloon were spherical, and a minor depression was made in its surface somewhere, it would destroy the symmetry and hence create non-zero field at all points.

Is there a better logical proof for it? Not too advanced mathematics, but i can understand integration (for example i understand the Gauss's Law) and am willing to learn a few more concepts, if necessary. A mathematical proof, not very advanced, would be a delight to me.

Thanks!