Stochastic Analysis for Poisson Point Processes by Giovanni Peccati & Matthias Reitzner

Author:Giovanni Peccati & Matthias Reitzner
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

thus relating the distributions of the number of vertices and the μ t -measure of Π t .

3.5.1 General Inequalities

Assume that is a compact convex set and set μ t (⋅ ) = tV d (K ∩⋅ ). We denote by Π t K  = conv[η t ] the Poisson polytope in K.

In this section we describe some inequalities for Poisson polytopes. Based on the work of Blaschke [11], Dalla and Larman [28], Giannopoulos [33], and Groemer [36, 37] showed that

(16)

where Π t △, resp. Π t B denotes the Poisson polytope where the underlying convex set is a simplex, resp. a ball of the same volume as K. The left inequality is true in arbitrary dimensions, whereas the right inequality is just known in dimension d = 2 and open in higher dimensions. To prove this extremal property of the simplex in arbitrary dimensions seems to be very difficult and is still a challenging open problem. A positive solution to this problem would immediately imply a solution to the hyperplane conjecture, see Milman and Pajor [76].

There are some elementary questions concerning the monotonicity of functionals of Π t K . First, it is immediate that for all and ,

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