\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
A threshold Boolean function is a Boolean
function defined on $\{0,1\}^n$ whose On-vertices and Off-vertices are strictly separable by a hyperplane in ${\cal R}^n$. Threshold logic is the main source for study of threshold Boolean functions, while Boolean algebraic methods have been the classical tools to study these objects. \\ \ \ \ \ Recently, we have claimed there exists a purely geometric approach to these linearly separable Boolean functions.
The principal motivation to this claim is the fact that these functions are just linearly separable cubical complexes and their place is convex geometry and polytopes. Here, we present a brief overview of a few results justifying this new connection.
\end{document}