Valuation And Operation On Convex Bodies

This thesis focuses on the study of Mikowski valuations compatible with general linear transformations in a sense,which is an extension of the study on Minkowski valuations compatible with orthogonal projections,the so-called translation-projection covariant valuation to the Minkowski valuation.We proved that under natural conditions,monotone Minkowski valuations on the set of 1 or 2 dimensional convex bodies compatible with a linear transformation are certainly linear transformations,and so give a characterization of linear transformations as valuations on Euclidean 1 or 2 spaces.We introduced also the scale multiplication on the unit sphere of the Euclidean space and discussed its basic properties,in terms of which,the homothety of a set on the unit sphere is defined naturally.This work provides some basic tool for further study on the valuation theory of convex sets on unit sphere.Main contributions of this thesis are as following:(1)Explicit expressions of the Minkowski valuation compatible with linear transformations on K1.(2)Valuation characteristics of linear transformations onR2.(3)Scale multiplication of a real number and a point on the unit sphere and its basic properties.