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Geometric Aspects of Probability Theory and Mathematical by V.V. Buldygin, A.B. Kharazishvili

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

It is celebrated that modern arithmetic comprises many disci­ plines. between them crucial are: set thought, algebra, topology, geometry, sensible research, chance conception, the speculation of differential equations and a few others. in addition, each mathematical self-discipline comprises numerous huge sections within which particular difficulties are investigated and the corresponding approach is constructed. for instance, generally topology we've got the subsequent huge chap­ ters: the idea of compact extensions of topological areas, the idea of continuing mappings, cardinal-valued features of topological areas, the idea of set-valued (multi-valued) mappings, and so forth. smooth algebra is featured by way of the subsequent domain names: linear algebra, staff thought, the speculation of earrings, common algebras, lattice concept, type idea, and so forth. touching on sleek chance conception, we will be able to simply see that the clas­ sification of its domain names is far extra huge: degree idea on ab­ stract areas, Borel and cylindrical measures in infinite-dimensional vector areas, classical restrict theorems, ergodic thought, normal stochastic approaches, Markov strategies, stochastical equations, mathematical facts, informa­ tion thought and plenty of others.

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Example text

We are going to prove this beautiful result here. So, let us assume that 1) - 3) are fulfilled. Pass through the origin of Rn the half-lines in such a way that every li (i = 1, 2, ... , k) has direction ei and its end-point coincides with 0. Then fix a family of strictly positive reals and, for each natural i E [1, k], take the affine hyperplane Gi = Gi (hi) in Rn orthogonal to ei and such that its distance from the origin is equal to hi (hence the linear segment on li between 0 and Gi(hi) has length hi).

8. e. Theorem 4 of this section) logically implies the Brunn-Minkowski inequality. BRUNN-MINKOWSKI INEQUALITY 35 9. Find necessary and sufficient conditions under which the Anderson inequality (see Theorem 5) is reduced to the equality. 10. We recall that a boundary point x of a convex body X in a topological vector space E is a point of strong convexity of X if there exists a hyperplane G in E which is support for X and satisfies the relation G n X = {x}. A convex body in E is called strongly convex if all its boundary points are points of strong convexity.

But then the equality (v(M,P,P, ... ,P))n = v(M)(v(P))n- 1 must be true. On the other hand, as demonstrated in Section 2, the latter equality is satisfied if and only if the given polyhedra M and P are homothetic. Since these polyhedra have the same volume, we may assert that the homothcty coefficient is equal to 1 and therefore M turns out to be a translate of P. This establishes the uniqueness of a polyhedron possessing the properties described. In this way, we have proved the following classical theorem due to Minkowski (see [150]).

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