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 continuous, real-valued functions of variables, . Assume that for each variable , the function is nonpositive when and nonnegative when . Then there is a point in the -dimensional cube in which all functions are simultaneously equal to .

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

A graphical representation of Poincaré–Miranda theorem for n = 2
A graphical representation of Poincaré–Miranda theorem for n = 2

The picture on the right shows an illustration of the Poincaré–Miranda theorem for n = 2 functions. Consider a couple of functions (f,g) whose domain of definition is [-1,1]2 (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 0. 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 xi, let ai be any value in the range [supxi = 0 fi, infxi = 1 fi]. Then there is a point in the unit cube in which for all i:

.

This statement can be reduced to the original one by a simple translation of axes,

where

  • xi are the coordinates in the domain of the function
  • yi 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.

See also