Editing Intermediate value theorem (section)

=== 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 [[N-cube|cube]].

Vrahatis<ref>{{Cite journal |last=Vrahatis |first=Michael N. |date=2016-04-01 |title=Generalization of the Bolzano theorem for simplices |url=https://www.sciencedirect.com/science/article/pii/S0166864115005994 |journal=Topology and Its Applications |language=en |volume=202 |pages=40–46 |doi=10.1016/j.topol.2015.12.066 |issn=0166-8641}}</ref> presents a similar generalization to triangles, or more generally, ''n''-dimensional [[Simplex|simplices]]. Let ''D<sup>n</sup>'' be an ''n''-dimensional simplex with ''n''+1 vertices denoted by ''v''<sub>0</sub>,...,''v<sub>n</sub>''. Let ''F''=(''f''<sub>1</sub>,...,''f<sub>n</sub>'') be a continuous function from ''D<sup>n</sup>'' to ''R<sup>n</sup>'', that never equals 0 on the boundary of ''D<sup>n</sup>''. Suppose ''F'' satisfies the following conditions:

* For all ''i'' in 1,...,''n'', the sign of ''f<sub>i</sub>''(''v<sub>i</sub>'') is opposite to the sign of ''f<sub>i</sub>''(''x'') for all points ''x'' on the face opposite to ''v<sub>i</sub>'';
* The sign-vector of ''f''<sub>1</sub>,...,''f<sub>n</sub>'' on ''v''<sub>0</sub> is not equal to the sign-vector of ''f''<sub>1</sub>,...,''f<sub>n</sub>''  on all points on the face opposite to ''v<sub>0</sub>''.

Then there is a point ''z'' in the [[Interior (topology)|interior]] of ''D<sup>n</sup>'' on which ''F''(''z'')=(0,...,0).

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

*For all ''i'' in 1,...,''n'', ''f<sub>i</sub>''(''v<sub>i</sub>'')>0, and ''f<sub>i</sub>''(''x'')<0 for all points ''x'' on the face opposite to ''v<sub>i</sub>''. In particular, ''f<sub>i</sub>''(''v<sub>0</sub>'')<0.
*For all points ''x'' on the face opposite to ''v<sub>0</sub>'', ''f<sub>i</sub>''(''x'')>0 for at least one ''i'' in 1,...,''n.''
The theorem can be proved based on the [[Knaster–Kuratowski–Mazurkiewicz lemma]]. In can be used for approximations of fixed points and zeros.<ref>{{Cite journal |last=Vrahatis |first=Michael N. |date=2020-04-15 |title=Intermediate value theorem for simplices for simplicial approximation of fixed points and zeros |journal=Topology and Its Applications |language=en |volume=275 |pages=107036 |doi=10.1016/j.topol.2019.107036 |issn=0166-8641|doi-access=free }}</ref>