You asked how to prove a function is convex---but it looks like your real question is how to prove that your local minimum is global. There are plenty of non-convex functions whose local minima are also global, and sometimes, you can even prove it :-)
One option you might consider is to employ a branch-and-bound global optimization approach. That's a systematic way for exploring the entire region of interest. In theory this could be very expensive, but in practice, you can often prune away large fractions of the region of interest, saving you a lot of computation.
Here are some slides and lecture notes by Stephen Boyd and Jacob Mattingley on the topic. To use this, you not only need to be able to compute local minima of your function (something you currently have) but the ability to compute good lower bounds for the function over arbitrary regions.
If you think this approach may have merit for your application, give us a little more detail about your problem (by editing your question) and perhaps I can fill in some more detail.