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 ${\displaystyle n}$ continuous, real-valued functions of ${\displaystyle n}$ variables, ${\displaystyle f_{1},\ldots ,f_{n}\colon [-1,1]^{n}\to \mathbb {R} }$. Assume that for each variable ${\displaystyle x_{i}}$, the function ${\displaystyle f_{i}}$ is nonpositive when ${\displaystyle x_{i}=-1}$ and nonnegative 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][2]: 545 [3]} It is sometimes called the Miranda theorem or the Bolzano–Poincaré–Miranda theorem.^[4]

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).

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.

By using topological degree theory it is possible to prove yet another generalization.^[5] Poincare-Miranda was also generalized to infinite-dimensional spaces.^[6]

• The Steinhaus chessboard theorem is a discrete theorem that can be used to prove the Poincare-Miranda theorem.^[7]

• Alefeld, Götz; Frommer, Andreas; Heindl, Gerhard; Mayer, Jan (2004). "On the existence theorems of Kantorovich, Miranda and Borsuk". ETNA. Electronic Transactions on Numerical Analysis [electronic only]. 18: 102–111.