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 $n$ functions in $n$ dimensions. It says as follows:


 * Consider $$n$$ continuous, real-valued functions of $$n$$ variables, $$f_1,\ldots, f_n:[-1,1]^n\to \mathbb R$$. 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 $$[-1,1]^n$$ 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 $n = 2$ functions. Consider a couple of functions $n = 2$ whose domain of definition is $(f,g)$ (i.e., the unit square). The function $f$ is negative on the left boundary and positive on the right boundary (green sides of the square), while the function $g$ 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 $f$ is $[-1,1]2$. Therefore, there must be a "wall" separating the left from the right, along which $f$ is $0$ (green curve inside the square). Similarly, there must be a "wall" separating the top from the bottom, along which $g$ is $0$ (red curve inside the square). These walls must intersect in a point in which both functions are $0$ (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 $0$, let $xi$ be any value in the range $ai$. Then there is a point in the unit cube in which for all $i$:
 * $$f_i=a_i$$.

This statement can be reduced to the original one by a simple translation of axes,
 * $$x^\prime_i=x_i\qquad y^\prime_i=y_i-a_i\qquad \forall i\in\{1,\dots,n\}$$

where By using topological degree theory it is possible to prove yet another generalization. Poincare-Miranda was also generalized to infinite-dimensional spaces.
 * $[supundefined fi, infundefined fi]$ are the coordinates in the domain of the function
 * $xi$ are the coordinates in the codomain of the function.