We define the excess degree $\xi(P)$ of a $d$-polytope $P$ as $2f_1-df_0$, where $f_0$ and $f_1$ denote the number of vertices and edges, respectively. This parameter measures how much $P$ deviates from being simple. It turns out that the excess degree of a $d$-polytope does not take every natural number: the smallest possible values are $0$ and $d-2$, and the value $d-1$ only occurs when $d=3$ or 5. On the other hand, for fixed $d$, the number of values not taken by the excess degree is finite if $d$ is odd, and the number of even values not taken by the excess degree is finite if $d$ is even. The excess degree is then applied in three different settings. First, it is used to show that polytopes with small excess (i.e., $\xi(P)

When designing wireless communication systems, it is very important to know the optimum numbers and locations for the access points (APs). The impact of incorrect placement of APs is significant. Placing too many access points increases the cost of deployment and interference, while placing too few access points can lead to coverage gaps. In this paper we describe a novel mathematical model developed to find the optimal number of APs and their locations in an environment that includes obstacles. To solve the problem, we use a new global optimization (AGOP) algorithm. The results obtained indicate that our model and software is able to solve optimal coverage problems for a design area with different number of users