The HamSandwich and the Brouwer Fixed Point Theorem

The Ham-Sandwich and the Brouwer Fixed Point Theorem Mathematics Seminar 14 September 2016 Mahidol University International College Wayne Lawton Adjunct Professor School of Mathematics and Statistics University of Western Australia, Perth, Australia wlawton 50@gmail. com https: //sites. google. com/site/wlawton 2011

Intermediate Value Theorem Bolzano, B. , 1817. Rein analytischer Beweis des Lehrsatzes, daß zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liegt. Gottlieb Haase, Prague.

Graphical Illustration

Uniform Continuity Lebesgue, H. (1909). "Sur les intégrales singulières". Ann. Fac. Sci. Univ. Toulouse. 3. pp. 25– 117.

Heine-Cantor Theorem https: //en. wikipedia. org/wiki/Heine%E 2%80%93 Cantor_theorem E. Heine, "Die Elemente der Funktionenlehre" J. Reine Angew. Math. , 74 (1872) pp. 172– 188 G. Cantor, E. Zermelo (ed. ) , Gesammelte Abhandlungen , Springer (1932)

Algorithmic Solution

Algorithmic Solution Theorem q Exists Proof Let We may assume that Since is odd hence nonzero. Let

Algorithmic Solution Theorem q Exists Proof Let We may assume that Since is odd hence nonzero. Let

Algorithmic Solution Pour-El and Richards showed that the intermediate value theorem is not computable because there is no algorithm that given a program for continuous computes a zero for The problem is the inverse m. o. c.

The Pancake Problem Cut two pancakes, each lying in a common plane, into equal amounts by a line L in the plane.

The Pancake Solution For each angle choose an of angle with oriented line and continuous function of Pancake 1 Pancake 2

The Pancake Solution Define continuous by Pancake 2 Pancake 1 so the result follows Clearly from the intermediate value theorem.

The Ham-Sandwich Theorem Three objects, consisting of a chunk of ham and two chunks of bread, can simultaneously be bisected with a single planar cut. It was proposed by Hugo Steinhaus and proved by Stefan Banach (explicitly in dimension 3, without bothering to automatically state theorem in the n-dimensional case), and also years later called the Stone–Tukey theorem after Arthur H. Stone and John Tukey.

The Ham-Sandwich Reduction Fix a closed hemisphere H of radius 1 centered at an origin and for point choose an oriented plane that bisects the ham and is a continuous function of Construct continuous by to reduce problem to the following version of Borsuk-Ulam Theorem continuous,

The Brouwer Fixed Point Theorem