The convex hull of a sample is used to approximate the support of the underlying distribution. This approximation has many practical implications in real life. For example, approximating the boundary of a finite set is used by many authors in environmental studies and medical research. To approximate the functionals of convex hulls, asymptotic theory plays a crucial role. Unfortunately, the asymptotic results are mostly very complicated. To address this complication, we suggest a consistent bootstrapping scheme for certain cases. Our resampling technique is used for both semi-parametric and non-parametric cases. Let X1,X2,...,Xn be a sequence of i.i.d. random points uniformly distributed on an unknown convex set. Our bootstrapping scheme relies on resampling uniformly from the convex hull of X1,X2,...,Xn. In this thesis, we study the asymptotic consistency of certain functionals of convex hulls. In particular, we apply our bootstrapping technique to the Hausdorff distance between the actual convex set and its estimator. We also provide a conjecture for the application of our bootstrapping scheme to Gaussian polytopes. Moreover, some other relevant consistency results for the regular bootstrap are developed.

This thesis is based on three papers on selected topics in Asymptotic Geometric Analysis. The first paper is about the volume of high-dimensional random polytopes; in particular, on polytopes generated by Gaussian random vectors. We consider the question of how many random vertices (or facets) should be sampled in order for such a polytope to capture significant volume. Various criteria for what exactly it means to capture significant volume are discussed. We also study similar problems for random polytopes generated by points on the Euclidean sphere. The second paper is about volume distribution in convex bodies. The first main result is about convex bodies that are (i) symmetric with respect to each of the coordinate hyperplanes and (ii) in isotropic position. We prove that most linear functionals acting on such bodies exhibit super-Gaussian tail-decay. Using known facts about the mean-width of such bodies, we then deduce strong lower bounds for the volume of certain caps. We also prove a converse statement. Namely, if an arbitrary isotropic convex body (not necessarily satisfying the symmetry assumption (i)) exhibits similar cap-behavior, then one can bound its mean-width. The third paper is about random polytopes generated by sampling points according to multiple log-concave probability measures. We prove related estimates for random determinants and give applications to several geometric inequalities; these include estimates on the volume-radius of random zonotopes and Hadamard's inequality for random matrices. / Mathematics

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The thesis deals with stochastic and algebraic aspects of the integer convex hull. In the first part, the intrinsic volumes of the randomized integer convex hull are investigated. In particular, we obtained an exact asymptotic order of the expected intrinsic volumes difference in a smooth convex body and a tight inequality for the expected mean width difference. In the algebraic part, an exact formula for the Bhattacharya function of complete primary monomial ideas in two variables is given. As a consequence, we derive an effective characterization for complete monomial ideals in two variables.