Poincaré–Miranda theorem

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

Consider ${\displaystyle n}$ continuous functions of ${\displaystyle n}$ variables, ${\displaystyle f_{1},\ldots ,f_{n}}$. Assume that for each variable ${\displaystyle x_{i}}$, the function ${\displaystyle f_{i}}$ is constantly negative when ${\displaystyle x_{i}=-1}$ and constantly positive when ${\displaystyle x_{i}=1}$. Then there is a point in the ${\displaystyle n}$-dimensional cube ${\displaystyle [-1,1]^{n}}$ in which all functions are simultaneously equal to ${\displaystyle 0}$.

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.[1]

Intuitive description

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 the [-1,+1]² 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 x_i, let a_i be any value in the range [sup_xi = 0 f_i, inf_xi = 1 f_i]. Then there is a point in the unit cube in which for all i:

${\displaystyle f_{i}=a_{i}}$.

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

${\displaystyle x_{i}^{\prime }=x_{i}\qquad y_{i}^{\prime }=y_{i}-a_{i}\qquad \forall i\in \{1,\dots ,n\}}$

where

• x_i are the coordinates in the domain of the function
• y_i are the coordinates in the codomain of the function.

Notes

1. (Kulpa 1997, p. 545).

References

• Dugundji, James; Granas, Andrzej (2003), Fixed Point Theory, Springer Monographs in Mathematics, New York: Springer-Verlag, pp. xv+690, ISBN 0-387-00173-5, MR 1987179, Zbl 1025.47002
• Kulpa, Wladyslaw (June 1997), "The Poincare-Miranda Theorem", The American Mathematical Monthly, 104 (6): 545–550, doi:10.2307/2975081, JSTOR 2975081, MR 1453657, Zbl 0891.47040.
• Miranda, Carlo (1940), "Un'osservazione su un teorema di Brouwer", Bollettino dell'Unione Matematica Italiana, Serie 2 (in Italian), 3: 5–7, JFM 66.0217.01, MR 0004775, Zbl 0024.02203.

External links

• Ahlbach, Connor Thomas (2013). "A Discrete Approach to the Poincare–Miranda Theorem (HMC Senior Theses)". Retrieved 18 May 2015.