James CollianderDepartment of Mathematics, University of TorontoGigliola StaffilaniDepartment of Mathematics, Massachusetts Institute of TechnologyMarkus KeelDepartment of Mathematics, University of Minnesota Twin CitiesHideo TakaokaDepartment of Mathematics Faculty of Science, Kobe UniversityTerence TaoDepartment of Mathematics, University of California
David DrasinDepartment of Mathematics, Purdue UniversityPekka PankkaDepartment of Mathematics and Statistics, P.O. Box 68, (Gustaf Hällströmin katu 2b), University of Helsinki, Finland
We show that given $${n \geqslant 3}$$ , $${q \geqslant 1}$$ , and a finite set $${\{y_1, \ldots, y_q \}}$$ in $${\mathbb{R}^n}$$ there exists a quasiregular mapping $${\mathbb{R}^n\to \mathbb{R}^n}$$ omitting exactly points $${y_1, \ldots, y_q}$$ .
The relationship between Lp affine surface area and curvature measures is investigated. As a result, a new representation of the existing notion of Lp affine surface area depending only on curvature measures is derived. Direct proofs of the equivalence between this new representation and those previously known are provided. The proofs show that the new representation is, in a sense, “polar” to that of Lutwak’s and “dual” to that of Schutt & Werner’s.
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