Efficient Algorithm for Proving Arbitrary Region in Higher-Dimensional Space is Bounded

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I have a region in n-dimensional space. I know nothing about this region, save these things:

  • To start, I am given the coordinates of a single point that I know is within the region
  • The region is contiguous
  • I can perform a computation on any point to determine whether or not it's also inside this region

By utilizing #3, what is the most efficient algorithm for proving, to a given sampling resolution, that my high-dimensional region is bounded?

(The region I am personally studying lies on a hypertorus, but I will also accept an answer that just applies to normal Euclidean hyperspace.)