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This is an extended version of a talk on October 4, 2004 at the research seminar " Differential geometry and applications " (headed by Academician A. T. Fomenko) at Moscow State University. The paper contains an overview of available (but far from well-known) results about the Blaschke addition of convex bodies, some new theorems on the monotonicity of the… (More)
A trivial formalization is given for the informal reasonings presented in a series of papers by Ya. D. Sergeyev on a positional numeral system with an infinitely large base, grossone; the system which is groundlessly opposed by its originator to the classical nonstandard analysis.
Let X be a Riesz space, and let Y be a Kantorovich (or Dedekind complete Riesz) space Y with base a complete Boolean algebra B. In  we have described every order bounded operator T from X to Y that may be presented as the difference of some Riesz homomorphisms. An operator T is such a difference if and only if the kernel of the stratum bT of T is a Riesz… (More)
This is a short overview of the life of Leonid Kantorovich and his contribution to the formation of the modern outlook on the interaction between mathematics and economics. Kantorovich was born in the family of a venereologist at St. Petersburg on January 19, 1912 (January 6, according to the old Russian style). It is curious that many reference books give… (More)
This is an overview of merging the techniques of Riesz space theory and convex geometry. Alexandr Danilovich Alexandrov became the first and foremost Russian geome-ter of the twentieth century. He contributed to mathematics under the slogan: " Retreat to Euclid, " remarking that " the pathos of contemporary mathematics is the return to Ancient Greece. "… (More)
The functional-analytical approach b y A. D. Alexandrov is discussed to the Minkowski and Blaschke structures making the set of convex compact gures into a vector space. The resulting analytical possibilities are illustrated by the isoperimetric type problems of nding convex gures separated by current h yperplanes similar to the Urysohn and double bubble… (More)
Dedicated to S. S. Kutateladze on occasion of his 60th birthday Recently, some new results on asymptotic behaviour of positive operators in Banach lattices were obtained. Here we discuss some open problems related to these results.
The main nonstandard tool-kits are known as innnitesimal analysis Robinson's nonstandard analysis and Boolean-valued analysis. Sharp distinctions between these two versions of nonstandard analysis in content and technique notwithstanding, many w a ys are open to their simultaneous application. One of the simplest approaches consists in successive… (More)
Under study is the new class of geometrical extremal problems in which it is required to achieve the best result in the presence of conflicting goals; e. g., given the surface area of a convex body x, we try to maximize the volume of x and minimize the width of x simultaneously. These problems are addressed along the lines of multiple criteria decision… (More)
This is a brief overview of the so-called " Case of Academician Luzin. " Tsunami swept over the Russian mathematical community in 1999 after publication of the book The Case of Academician Nikola˘ ı Nikolaevich Luzin . For the first time it revealed the complete shorthand notes of the meetings of a notorious special Commission of the Academy of Sciences… (More)