Two convex bodies K and K′ in Euclidean space [ ]n can be said to be in exceptional relative position if they have a common boundary point at which the linear hulls of their normal cones have a non-trivial intersection. It is proved that the set of rigid motions g for which K and gK′ are in exceptional relative position is of Haar measure zero. A similar result holds true if ‘exceptional relative position’ is defined via common supporting hyperplanes. Both results were conjectured by S. Glasauer; they have applications in integral geometry.

A new coupling of one-dimensional random walks is described which tries to control the coupling by keeping the separation of the two random walks of constant sign. It turns out that among such monotone couplings there is an optimal one-step coupling which maximises the second moment of the difference (assuming this is finite), and this coupling is ‘fast’ in the sense that for a random walk with a unimodal step distribution the coupling time achieved by using the new coupling at each step is stochastically no larger than any other coupling. This is applied to the case of symmetric unimodal distributions.