Continuous rotation invariant valuations on convex sets

The famous Hadwiger theorem classifies all rigid motion invariant continuous valuations on convex sets as linear conbinations of quermassintegrals. We prove much more general result. We classify continuous valuations which are invariant with respect to the orthogonal (or special orthogonal) group. Some applications to integral geometry are given.Comment: 29 pages, published version, abstract added in migratio

The p-adic local monodromy theorem for fake annuli

We establish a generalization of the p-adic local monodromy theorem (of Andre, Mebkhout, and the author) in which differential equations on rigid analytic annuli are replaced by differential equations on so-called fake annuli. The latter correspond loosely to completions of a Laurent polynomial ring with respect to a monomial valuation. The result represents a step towards a higher-dimensional version of the p-adic local monodromy theorem (the "problem of semistable reduction"); it can also be viewed as a slightly novel presentation of the original p-adic local monodromy theorem.Comment: 38 pages; v4: refereed version, some typos fixe