Intermediate value theorem
Multi-dimensional spaces

The Poincaré-Miranda theorem is a generalization of the Intermediate value theorem from a (one-dimensional) interval to a (two-dimensional) rectangle, or more generally, to an n -dimensional cube.

Vrahatis presents a similar generalization to triangles, or more generally, n -dimensional simplices. Let D be an n -dimensional simplex with n +1 vertices denoted by v,..., v. Let F =(f,..., f) be a continuous function from D to R, that never equals 0 on the boundary of D. Suppose F satisfies the following conditions:

Then there is a point z in the interior of D on which F (z)=(0,...,0).

It is possible to normalize the f such that f (v)>0 for all i; then the conditions become simpler:

The theorem can be proved based on the Knaster–Kuratowski–Mazurkiewicz lemma. In can be used for approximations of fixed points and zeros.
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