The book is devoted to beautiful geometric theorems of linear functional analysis, namely the theorems of Crane - Milman, Shock and other authors. Starting with simple geometric facts, the author gives transparent proofs of the mentioned theorems and leads a number of their applications to analysis - in the theory of Banach algebras (Shilov boundary), measure theory, integral representations of important classes of functions, inversion formulas, etc. Although the main facts of the presented theory have now become quite classical, there has not yet been such a clear, clear and elementary presentation, accompanied by a rich selection of non-trivial examples.