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In this paper we discuss the change in contact structures as their supporting open book decompositions have their binding components cabled. To facilitate this and applications we define the notion of a rational open book decomposition that generalizes the standard notion of open book decomposition and allows one to more easily study surgeries on transverse knots. As a corollary to our investigation we are able to show there are Stein fillable contact structures supported by open books whose monodromies cannot be written as a product of positive Dehn twists. We also exhibit several monoids in the mapping class group of a surface that have contact geometric significance.

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A longstanding question in the dual Brunn–Minkowski theory is “What are the dual analogues of Federer’s curvature measures for convex bodies?” The answer to this is provided. This leads naturally to dual versions of Minkowski-type problems: What are necessary and sufficient conditions for a Borel measure to be a dual curvature measure of a convex body? Sufficient conditions, involving measure concentration, are established for the existence of solutions to these problems.