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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 n continuous functions of n variables. Assume that for every variable \( x_i \), the function \( f_i \) is constantly negative when \( x_i = 0 \) and constantly positive when \( x_i = 1 \). Then there is a point in the n-dimensional unit cube 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.[1]
Intuition
PoincareMiranda.png
The figure on the right shows an illustration of the Poincaré–Miranda theorem for n=2 functions. The function f is negative on the left boundary and positive on the right boundary. 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). Similarly, there must be a "wall" separating the top from the bottom, along which g is 0 (red). These walls must intersect in a point in which both functions are 0 (blue).
Generalizations
For every variable \( x_i \), let \( a_i be any value in the range: \( [\sup_{x_i=0}{f_i},\inf_{x_i=1}{f_i}]\). Then there is a point in the unit cube in which for all \( i: f_i=a_i\). This can be reduced to the original statement by a simple translation of coordinates.
References
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Ahlbach, Connor Thomas (2013). "A Discrete Approach to the Poincare–Miranda Theorem (HMC Senior Theses)". Retrieved 18 May 2015.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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