Convex Geometry I. Definitions, Properties and Fundamental Theorems

Convex Geometry is the branch of geometry studying convex sets, mainly in Euclidean spaces. Convex sets occur naturally in Geometry and in many mathematical areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming, game theory, probability theory, stochastic geometry, stereology etc.. Convex Geometry is also of interest in other scientific and engineering disciplines (e.g. in biology, chemistry, cosmology, geology, pharmaceutics, physics …) where elementary objects (cells, corpuscles, grains, particles, planets …) are often considered as convex sets. This first article deals with the main definitions and properties and fundamental theorems relating to convex sets and more broadly to star-shaped sets.