Valuations on Convex Bodies: The Classical Basic Facts

  • Rolf SchneiderEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 2177)


The purpose of this chapter is to give an elementary introduction to valuations on convex bodies. The goal is to serve the newcomer to the field, by presenting basic notions and collecting fundamental facts, which have proved of importance for the later development, either as technical tools or as models and incentives for widening and deepening the theory. We also provide hints to the literature where proofs can be found. It is not our intention to duplicate the existing longer surveys on valuations, nor to update them. We restrict ourselves to classical basic facts and geometric approaches, which also means that we do not try to describe the exciting developments of valuation theory in the last 15 years, which involve deeper methods and will be the subject of later chapters. The sections of the present chapter treat, in varying detail, general valuations, valuations on polytopes, examples of valuations from convex geometry, continuous valuations on convex bodies, measure-valued valuations, valuations on lattice polytopes.


Convex Body Rigid Motion Characterization Theorem Intrinsic Volume Outer Normal Vector 
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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Mathematical InstituteAlbert-Ludwigs-Universität FreiburgFreiburgGermany

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