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).
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