From L_p Minkowski Valuations To Orlicz Valuations And To General Functions Valued Valuations

The researches of this thesis is belong to valuations theory. Valuations theory is a subject that combined operator theory and convex geometry analysis. Convex geometry analysis is an important branch of modern geometry analysis. It has many applications on analysis, differential geometry, integral geometry, PDE, information theory, stochastic geometry, locally Banach space theory, etc.. Valuations theory begins due to Dehn’s solution of Hilbert’s third problem. The systematic study of valuations is from the found of the Hadwiger theorem, which become a valuable tool for generating quick and effortless proofs of many results for invariant geometry operators.The develop of valuations theory and a summary of our works are introduced in the first chapter of this thesis. In the second chapter, we collect some preliminaries and notations. In the third chapter, Orlicz valuations are studied. We show that SL(n) compatible (covariant or contravariant) Orlicz valuations have non trivial solutions if and only if they are Lp Minkowski valuations. In forth chapter, Lp Minkowski valuations are studied. We completely classify SL(n) compatible Lp Minkowski valuations, which are classified before by Ludwig, Haberl, and Parapatits with some additional conditions. Besides, we also classify SL(n) compatible L∞ Minkowski valuations. The classification of Lp Minkowski valuations is based on classification of continuous p homogeneous functions valued valuations. In the fifth chapter, another functions valued valuations are studied. It lead us to characterize Laplace transforms.