Math-Selfie

This is a write up on some sections of convex geometry, functional analysis, optimization, and nonstandard models that attract the author.

💡 Research Summary

The manuscript entitled “Math‑Selfie” is a personal survey of four interrelated research areas that the author, S.S. Kutateladze, has pursued over several decades: convex geometry, functional analysis, optimization theory, and non‑standard models of set theory. The paper is organized into four thematic sections, each presented in a roughly chronological order.

1. Optimal Location of Convex Bodies – The author revisits classical isoperimetric and Urysohn-type problems, showing how the modern linear‑programming duality framework (originating with L.V. Kantorovich) can be combined with mixed‑volume theory (H. Minkowski) and measure‑theoretic constructions (Y.G. Reshetnyak) to treat extremal placement problems that are not amenable to symmetry arguments. By translating these geometric extremal problems into convex programs in appropriate function spaces, new classes of inequalities for convex surfaces are obtained. The paper highlights the “soup‑bubble” solution inside a tetrahedron, which can be described as the vector sum of a ball and the solution of an internal Urysohn problem, and mentions recent work on double‑bubble configurations.

2. Ordered Vector Spaces and K‑Spaces – This section focuses on Kantorovich’s heuristic principle and the development of K‑spaces (Dedekind‑complete vector lattices). The author explains how the Hahn–Banach extension theorem can be abstracted by replacing real scalars with elements of an arbitrary K‑space, leading to the so‑called identity‑preservation theorems. The “transfer principle” is given a modern interpretation via non‑standard set‑theoretic models, showing that K‑spaces can be regarded as dense subfields of the reals within a suitable non‑standard universe. Applications to mathematical economics are discussed, especially the role of K‑spaces in modeling divisible goods and in extending linear programming beyond rational coefficients.

3. Nonsmooth Analysis and Approximate Optimization – Here the author develops a systematic approach to ε‑programming and infinitesimal analysis. By treating the error vector ε as an infinitesimal, one can formulate approximate solutions to vector‑valued optimization problems where classical differential calculus fails. The paper introduces the “Kutateladze canonical operator” and associated approximate solution concepts, which allow the reduction of multi‑criteria convex programs to explicit surface‑measure conditions. The author also revisits Urysohn‑type problems with additional targets (flattening, symmetry, convex hull volume) and shows how the new sub‑differential calculus yields compact necessary‑optimality conditions.

4. New Models for Mathematical Analysis – The final part surveys the author’s work on non‑standard and Boolean‑valued models. Two main strands are identified: (a) infinitesimal analysis à la Robinson, used primarily to simplify definitions and proofs, and (b) Boolean‑valued analysis, which introduces a richer syntax based on the predicate of “standardness.” The author describes the construction of non‑standard hulls, Loeb measures, hyper‑approximation, and cyclic monads, emphasizing their utility in functional analysis, the theory of von Neumann algebras, and vector measure theory. The paper argues that Boolean‑valued universes provide a natural setting for universally complete vector lattices (extended K‑spaces), effectively serving as alternative models of the real line and thereby extending Kantorovich’s heuristic principle.

Overall, the manuscript weaves together classical convex‑geometric extremal problems, the algebraic‑order structure of K‑spaces, modern nonsmooth optimization techniques, and sophisticated logical frameworks (non‑standard and Boolean‑valued models). It demonstrates how these disparate tools can be combined to obtain new results in geometry, optimization, and economic theory, and it positions the author’s contributions within the broader Russian mathematical tradition while pointing toward future interdisciplinary applications.