Density-functional theory (DFT) is a computational modelling tool used to describe molecules and materials. Different functions are used to determine the properties of electrons and molecules in solids. It is the most widely used method in electronic structure calculations in chemistry, material sciences and physics. Density-Functional Theory: A Convex Treatment gives an introduction to the more mathematical aspects of density-functional theory, allowing a larger group of theoretical chemists and physicists to obtain a full understanding of the theoretical foundation of DFT. Relevant mathematical apparatus, including functional and convex analysis, are introduced and developed before being applied in the subsequent chapter, allowing readers to develop their foundation of DFT. Recent mathematical developments which allow the simplifications of many original proofs while providing significant new insights, are also presented. Topics covered include: -Hohenberg-Kohn theory -Vector spaces and linear functionals -Convex sets and their separation -Lieb constrained-search theory -Convex conjugation and duality -Grand canonical ensembles -Thomas-Fermi theory -The adiabatic connection -Scaling relations Exercises and detailed solutions can be found throughout the book. Density-Functional Theory: A Convex Treatment will provide a consistent and focused description of the fundamentals of DFT, making the important fundamental facts about DFT more accessible to graduate students in electronic structure theory, researchers in chemistry, physics, and materials science as well as theoretical chemists.