Convex bodies all whose sections (projections) are equal

The purpose of this paper is to answer the following question: If all hyperplane sections through the origin of a convex body are "equal", is the convex body "equal" to the ball? The meaning of the notion "equal" will change in the course of this paper. Similarly, we are interested in the following problem: If all orthogonal projections of a convex body onto hyperplanes are "equal", is the convex body "equal" to the ball? Topology and convex geometry are deeply interrelated in the solution and understanding of these problems.