Let \(\gamma^n_k\) be the tautological bundle over the Grassmannian of \(k\)-dimensional linear subspaces of \({\mathbb R}^n\). A field of convex bodies (f.c.b.) in \(\gamma^n_k\) associates with each fiber of the bundle a \(k\)-dimensional convex body in the fiber that depends continuously on the fiber. (Examples are provided by the \(k\)-dimensional planar sections of a convex body \(K\subset {\mathbb R}^n\) through a fixed interior point, or by the orthogonal projections of \(K\) to the \(k\)-dimensional subspaces.)
This paper collects numerous examples, often deduced from earlier results of the author, of the existence of convex bodies in every f.c.b. that have qualitatively better properties than general \(k\)-dimensional convex bodies. Typical examples: Every f.c.b. in \(\gamma^3_2\) contains a convex body \(K\) such that in its fiber there are homothetic ellipses \(E_1,E_2\) with the same center and with dilatation factor \(\sqrt{2}\) such that \(E_1\subset K\subset E_2\) (for comparison: dilatation factor \(2\) for general \(K\)). Every f.c.b. in \(\gamma^3_2\) contains a convex body \(K\) that admits a lattice packing in the plane with density at least \(3/4\) (for comparison: density at least \(2/3\) for general \(K\)). The topics treated in this spirit comprise symmetry measures, mass partitions, lengths of unit circles in normed planes, universal covers, Jung’s theorem, and others. Further results concern polytopes, for example: Every f.c.b. in \(\gamma^n_k\) consisting of polytopes contains a polytope with at least \(n-k+2\) facets (vertices).