By P. M. Gruber, J. M. Wills, Arjen Sevenster

The guide has 3 goals. One is to survey, for specialists, convex geometry in its ramifications and its family members with different parts of arithmetic. A moment target is to provide destiny researchers in convex geometry a high-level advent to such a lot branches of convexity and its purposes, exhibiting the main rules, tools, and effects; The 3rd objective is to turn out invaluable for mathematicians operating in different parts, in addition to for econometrists, computing device scientists, crystallographers, physicists, and engineers who're trying to find geometric instruments for his or her personal paintings. quantity B covers discrete, analytic, and stochastic points of convexity.

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Let Z1 ; Z2 ; : : : be independent observations from the distribution of Z. Choose yn so that P fY yn g 1=n. Z H / ) W as above. J / < 1: (8) n D n˛n . / ! The restriction of to HC is a probability measure, the distribution of W . 6. It is a central result. The first limit relation is analytical. It raises questions such as: 1) What limit laws are possible? 2) For a given limit law, what conditions on the distribution of Z will yield convergence? Preview 19 The second relation is more geometric.

1; (22) where Z r is the vector Z conditioned to lie outside the open ball rB. In this situation it is natural to use polar coordinates and write Z D R with R D kZk. The distribution of . ; R=r/, conditional on R r, converges to a product measure d dG on 26 Preview @B Œ1; 1/, where is the spectral measure, and G a Pareto distribution on Œ1; 1/ with density =r C1 , D 1= . The spectral measure gives an idea of the directions in which the data extremes cluster; the parameter in (21) describes the decay rate of the tails.

Our theory is based on concepts like scale invariance, self-similarity and symmetry. It is geometric and local. Independence is a global analytic assumption. It allows one to draw farreaching conclusions about extremes, but the techniques are different from those developed here. B/. These situations yield a limit measure with a one-parameter family of symmetries, the excess measure described in (9). Let us now return to high risk scenarios Z H where the halfspaces H are allowed to diverge in any direction.