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Si us plau utilitzeu sempre aquest identificador per citar o enllaçar aquest document: https://hdl.handle.net/2445/121133
On the proof of the upper bound theorem
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[en] Let $\Delta$ be a triangulation of a $(d - 1)$-dimensional sphere with $n$ vertices. The Upper Bound Conjecture (UBC for short) gives an explicit bound of the number of $i$-dimensional faces of $\Delta$.
This question dates back to the beginning of the 1950’s, when the study of the efficiency of some linear programming techniques led to the following problem: Determine the maximal possible number of $i$-faces of d-polytope with $n$ vertices.
The first statement of the UBC was formulated in 1957 by Theodore Motzkin. The original result state that the number of $i$-dimensional faces of a $d$-dimensional polytope with n vertices are bound by a certain explicit number $f i (C(n, d))$ where $C(n, d)$ is a cyclic polytope and $f_{i}$ denotes the number of $i$-dimensional faces of the simplex. We say that $P$ is a polytope if it is the convex hull of a finite set of points in $\mathbb{R}^{d}$. Moreover, we say that $C(n, d)$ is a cyclic polytope if it is the convex hull of n distinct points on the moment curve $(t, t^{2},..., t{^d})$, $-\infty
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Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2017, Director: Santiago Zarzuela
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DEDIU, Catalin. On the proof of the upper bound theorem. [consulted: 15 of August of 2026]. Available at: https://hdl.handle.net/2445/121133