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Identifier

uch.math.phd//2003chartzoulaki

Title

Πιθανοθεωρητικές μέθοδοι στη θεωρίσ των κυρτών σωμάτων

Creator

Chartzoulaki, Marianna

Abstract

The main theme of this Ph.D. Thesis is the use of probabilistic methods in the theory of high-dimensional convex bodies. We discuss the following aspects of the theory: 1. Volume ratio. Let K and L be two convex bodies in R^n. The volume ratio vr(K,L) of K and L is defined by vr(K,L)=inf( |K|/|T(L)|)^{1/n}, where the infimum is over all affine transformations T of R^n for which T(L)<=K. We show that vr(K,L)<=c sqrt{n}log n, where c>0 is an absolute constant. This is optimal up to the logarithmic term. We also discuss the dimension of spherical sections of symmetric convex bodies K in R^n with bounded volume ratio vr(K,B_2^n) (possible improvements of Szarek's theorem). 2. 0-1 polytopes. Let E_2^n={ -1,1}^n be the discrete cube in R^n. For every N>=n we consider the class of convex bodies K_N= co{+or- x_1,...,+or- x_N} which are generated by N random points x_1,...,x_N chosen independently and uniformly from E_2^n. We show that if n>= n_0 and N>= n(log n)^2 then, for a random K_N, the inradius, the volume radius, the mean width and the size of the maximal inscribed cube can be determined up to an absolute constant as functions of $n$ and N . This geometric description of K_N leads to sharp estimates for several asymptotic parameters of the corresponding n-dimensional normed space X_N. 3. Random polytopes in a convex body. Let K be a convex body in R^n with volume |K|=1. We choose N>= n+1 points x_1,...,x_N independently and uniformly from K, and write C(x_1,... ,x_N) for their convex hull. Let f: R^+ -> R^+ be a continuous strictly increasing function and 0<= i<= n-1. Then, the quantity E(K,N,foW_i)=S_K\...S_Kf[W_i(C(x_1,...,x_N))]dx_N... dx_1 is minimal if K is a ball (W_i is the i-th quermassintegral of a compact convex set). If f is convex and strictly increasing and 1<= i<= n-1, then the ball is the only extremal body. These two facts generalize a result of H. Groemer on moments of the volume of C(x_1,... ,x_N). In the case of a 1-unconditional convex body, using recent results of Bobkov and Nazarov, we show that the volume radius F(K,N)=S_K...S_K|C(x_1,... ,x_N)|^{1/n}dx_N... dx_1 is of the order of sqrt{log (N/n)}/sqrt{n}} for all n(log n)^2<= N<=exp(cn).

Issue date

2003-03-01

Date available

2003-06-30

Collection

School/Department--School of Sciences and Engineering--Department of Mathematics--Doctoral theses

Type of Work--Doctoral theses

Permanent Link

https://elocus.lib.uoc.gr//dlib/d/e/2/metadata-dlib-2003chartzoulaki.tkl

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805

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