Poincaré–Miranda theorem

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

Consider $n$ continuous, real-valued functions of $n$ variables, $f_1,\dots ,f_n:[-1,1{]}^n\to ℝ$. 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.Template:Rp It is sometimes called the Miranda theorem or the Bolzano–Poincaré–Miranda theorem.

Intuitive description

Template:Plain image with caption The picture on the right shows an illustration of the Poincaré–Miranda theorem for Template:Math functions. Consider a couple of functions Template:Math whose domain of definition is Template:Math (i.e., the unit square). The function Template:Mvar is negative on the left boundary and positive on the right boundary (green sides of the square), while the function Template:Mvar 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 Template:Mvar is Template:Math. Therefore, there must be a "wall" separating the left from the right, along which Template:Mvar is Template:Math (green curve inside the square). Similarly, there must be a "wall" separating the top from the bottom, along which Template:Mvar is Template:Math (red curve inside the square). These walls must intersect in a point in which both functions are Template:Math (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 Template:Math, let Template:Math be any value in the range Template:Math. Then there is a point in the unit cube in which for all Template:Mvar:

$f_i=a_i$.

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

$x_i^{\prime }=x_i{y}_i^{\prime }=y_i-a_i\mathrm{\forall }i\in \{1,\dots ,n\}$

where

• Template:Math are the coordinates in the domain of the function
• Template:Math 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

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

Further reading

• Template:Cite journal