Poincaré–Miranda theorem

In mathematics, the Poincaré–Miranda theorem is a generalization of intermediate value theorem, from a single function in a single dimension, to functions in dimensions. It says as follows:

::Consider $n$ continuous functions of $n$ variables, $f_1,\ldots, f_n$. Assume that for each variable $x_i$, the function $f_i$ is nonpositive when $x_i=-1$ and nonnegative when $x_i=1$. Then there is a point in the $n$-dimensional cube 1,1n in which all functions are ''simultaneously'' equal to $0$.

The theorem is named after Henri Poincaré - who conjectured it in 1883, and Carlo Miranda - who in 1940 showed that it is equivalent to the Brouwer fixed-point theorem. It is sometimes called the Miranda theorem or the Bolzano-Poincare-Miranda theorem.

Intuitive description

The picture on the right shows an illustration of the Poincaré–Miranda theorem for functions. Consider a couple of functions whose domain of definition is (i.e., the unit square). The function is negative on the left boundary and positive on the right boundary (green sides of the square), while the function is negative on the lower boundary and positive on the upper boundary (red sides of the square). When we go from left to right along ''any'' path, we must go through a point in which is . Therefore, there must be a "wall" separating the left from the right, along which is (green curve inside the square). Similarly, there must be a "wall" separating the top from the bottom, along which is (red curve inside the square). These walls must intersect in a point in which both functions are (blue point inside the square).

Generalizations

The simplest generalization, as a matter of fact a corollary, of this theorem is the following one. For every variable , let be any value in the range .
Then there is a point in the unit cube in which for all :
:$f_i=a_i$.

This statement can be reduced to the original one by a simple translation of axes,
:$x^\prime_i=x_i\qquad y^\prime_i=y_i-a_i\qquad \forall i\in\$
where
* are the coordinates in the domain of the function
* are the coordinates in the codomain of the function.
By using topological degree theory it is possible to prove yet another generalization. Poincare-Miranda was also generalized to infinite-dimensional spaces.

References

Further reading

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{{DEFAULTSORT:Poincare-Miranda theorem
Topology
Real analysis