Bonnesen-Style Inequalities for Minkowski Relative Geometry

Sangwine-Yager, J. R.

doi:10.2307/2000768

Cited by 11 publications

(4 citation statements)

References 7 publications

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“…Of course, in a general Banach space the inclusion A + XU C B\(A) may be proper; the sets A + XU are often called enlargements of A. Following standard practice in finite dimensions [7,14], if A is a closed convex body in a normed linear space X with A ^ X, we call the superlevel sets of the concave functional d( •, A c ) restricted to A inner parallel bodies of the set A. For notation, if A ^ 0, we write…”

Section: Sublevel Sets Of the Distance Functional D( • A)mentioning

confidence: 99%

Quasi-concave functions and convex convergence to infinity

Beer

1999

Bull. Austral. Math. Soc.

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By a convex mode of convergence to infinity {Ck), we mean a sequence of nonempty closed convex subsets of a normed linear space X such that for each k, oo Ck+i C intCfc and f] Ck -%, and a sequence (x n ) is X is declared convergent to infinity with respect to {Ck) provided each Ck contains x n eventually. Positive convergence to infinity with respect to a pointed cone with nonempty interior as well as convergence to infinity in a fixed direction fit within this framework. In this paper we study the representation of convex modes of convergence to infinity by quasi-concave functions and associated remetrizations of the space.

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Section: Sublevel Sets Of the Distance Functional D( • A)mentioning

confidence: 99%

Quasi-concave functions and convex convergence to infinity

Beer

1999

Bull. Austral. Math. Soc.

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“…Bokowski and Heil [4], Osserman [12] and J. R. Sangwine-Yager [16], [17]. There is a widespread literature on Bonnesen-style inequalities, cf.…”

Section: Thus (3) Becomesmentioning

confidence: 99%

On Blaschke's extension of Bonnesen's inequality

1993

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“…The study of inner parallel bodies is not only interesting but useful, since it is connected with other nice problems for compact convex sets. Wills , Sangwine‐Yager , Brannen and others proved many more general and stronger Bonnesen‐style inequalities by using the technique of inner parallel bodies. The classical Bonnesen‐style inequality was first given by Bonnesen himself in , and investigated in many papers (see e.g., ).…”

Section: Introductionmentioning

confidence: 99%