We deal with five different problems from convex geometry, each on its own chapter of this Thesis. These problems are the following. Random copies of a convex body: We study the probability that a random copy of a convex body intersects the integer lattice in a certain way. A conjecture by Erdos: We study the statement by Erdos "On every convex curve there exists a point P such that every circle with centre P intersects the curve in at most 2 points." A Yao-Yao type theorem: Given a nice measure in R^d, we show that there is a partition P of R^d into 3*2^(d/2) convex pieces of equal measure such that every hyperplane avoids at least 2 elements of P. Line transversals: Given a family F of balls in R^d such that every three have a transversal line, we bound the blow-up factor l needed so that lF has a line transversal. Longest lattice convex chains: Given a triangle with two specified vertices v_1, v_2 in Z^2, we bound the size of the largest lattice convex chain from v_1 to v_2. The techniques used to tackle these problems are very diverse and include results from analysis, combinatorics, number theory and topology, as well as the use of computers
