Spernerova lema i primjene

Njemački matematičar Emmanuel Sperner prvi je dokazao rezultat koji će se kasnije po njemu zvati Spernerova lema, a on kaze da svaka Spernerova triangulacija n-simpleksa sadrži neparan broj potpuno označenih n-simpleksa. U ovom radu navodimo dokaz tog rezultata, kao i njegovu primjenu na probleme pravedne raspodjele torte i stanarine. Osim toga, pokazujemo vezu s Brouwerovim teoremom o fiksnoj točki i na kraju navodimo vezu Brouwerovog teorema s igrom Hex.Emmanuel Sperner, a German mathematician, was the first to prove a result which will later be named after him and known as Sperner’s lemma, which says that every Sperner triangulation of an n-simplex contains an odd number of fully labelled n-simplexes. We present the proof of that result and apply it on fair division problems, namely on cake cutting and rental division. Besides that we show the connection with the Brouwer’s fixed point theorem and in the end the connection between the Brouwer’s theorem and the game of Hex