Let A be a compact subset of a (real) Hilbert space H. Denote by co(A) the convex hull of A. We will discuss the following problem: How much larger is co(A) than A? We present several bounds for the size of co(A), in dependence on that of A. Here the sizes of A and of co(A) are measured by their entropy numbers e_n(A) or, equivalently, by their covering numbers N(A, Iu). Roughly spoken, the results assert the following: If A is a**smalla**, then co(A) can be much bigger, but its size cannot exceed a certain critical bound. On the other side, if A is already a**biga**, then co(A) has about the same size as A. Of special interest is the critical case which corresponds to e_n(A) ~ n^{a**1/2} or, equivalently, to logN(A, Iu) ~ Iu^{a**2}. No geometrical explanation for the existence of this critical case is known. But there exists a probabilistic reason in terms of Gaussian random processes. We will discuss this relation in detail. Finally, if time admits, we present some applications of the results. They allow us to estimate the entropy numbers for certain summation operators on binary trees.