By V.V. Buldygin, A.B. Kharazishvili

It is celebrated that modern arithmetic contains many disci plines. between them crucial are: set idea, algebra, topology, geometry, useful research, likelihood thought, the idea of differential equations and a few others. additionally, each mathematical self-discipline comprises numerous huge sections during which particular difficulties are investigated and the corresponding procedure is constructed. for instance, more often than not topology we've the next broad chap ters: the idea of compact extensions of topological areas, the idea of constant mappings, cardinal-valued features of topological areas, the idea of set-valued (multi-valued) mappings, and so on. smooth algebra is featured through the next domain names: linear algebra, staff idea, the idea of jewelry, common algebras, lattice idea, type idea, etc. relating smooth likelihood idea, we will simply see that the clas sification of its domain names is far extra wide: degree idea on ab stract areas, Borel and cylindrical measures in infinite-dimensional vector areas, classical restrict theorems, ergodic thought, basic stochastic tactics, Markov procedures, stochastical equations, mathematical facts, informa tion idea and plenty of others.

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**Example text**

Note now that the natural algebraic operations in Rn possess canonical extensions to the class of all nonempty compact subsets of Rn. Indeed, let X and Y be any two compact subsets of R n and let a and b be any two real numbers. Then the set aX+ bY= {ax+ by : x EX, y E Y} is compact, too, because it coincides with the range of the continuous mappmg ( x, y) ---+ ax + by (xEX, yEY) defined on the compact product space X x Y. In addition, if the given sets X and Y are convex, then the set aX + bY is convex as well.

Hence (St( r) )1/(n-1) > (1 - t)( so( r) )1/(n-1) + t( sl ( r) )1/(n-1). So the preceding relations imply the inequality 1 1 v(Ct) > ((1- t)(s 0 (r)) 1 /(n- 1 ) + t(s 1 (r)) 1 /(n- 1))n- 1 ((1- t)/so(r) + t/s1(r))dr. Let us demonstrate that the function f under the integral above is always greater than or equal to 1. (~) = (1- t + t. e/(n-l)t- 1(1- t + tj~). Obviously, lim~-o+f(e) ~, is of = lim~-+oof(~) = +oo. Therefore, our f attains its infimum at some point of the interval ]0, +oo[. The differentiation of f easily leads to the expression 25 BRUNN-MINKOWSKI INEQUALITY = = This formula shows that f' (~) 0 if and only if ~ 1.

In particular, we have Theorem 1. The pair (Comp(Rn ), d) is a complete separable metric space. Besides, this space turns out to be locally compact. , [18]). Theorem 2. Let (Xi)iEI be an arbitrary family of nonempty compact subsets of Rn! such that all these sets are contained in some ball of Rn. Then there exists a countable subfamily of (Xi)iEI converging (in metric d) to some non empty compact subset of Rn. The result just formulated is usually called the Blaschke selection theorem. It follows at once from this result that any closed ball in the space of 19 20 BRUNN -MINKOWSKI INEQUALITY all nonempty compact subsets of Rn (equipped with the above-mentioned metric d) is a compact space.