Porosity and diametrical completeness

Moreno, J. P.

doi:10.1007/s11856-021-2151-z

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Definition 27 ([40]1

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2023

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“…It is obvious that a diametrically complete set has to be closed and convex. Many other interesting properties of this kind of sets can be found in [11,[36][37][38][39]41] and in the literature given therein.…”

Section: Definition 27 ([40]mentioning

confidence: 99%

Diametrically Complete Sets with Empty Interior and Constant Width Sets with Empty Interior

Kaczor¹,

Kuczumow²,

Reich

et al. 2023

Results Math

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We show that each infinite-dimensional reflexive Banach space $$(X,\left\| \cdot \right\| _X)$$ ( X , · X ) has an equivalent norm $$\left\| \cdot \right\| _{X,0}$$ · X , 0 such that $$(X,\left\| \cdot \right\| _{X,0})$$ ( X , · X , 0 ) is LUR and contains a diametrically complete set with empty interior. We also prove that after a suitable equivalent renorming, the Banach space $$C([0,1],{\mathbb {R}})$$ C ( [ 0 , 1 ] , R ) contains a constant width set with empty interior.

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Section: Definition 27 ([40]mentioning

confidence: 99%

Diametrically Complete Sets with Empty Interior and Constant Width Sets with Empty Interior

Kaczor¹,

Kuczumow²,

Reich

et al. 2023

Results Math

View full text Add to dashboard Cite

show abstract

Complete sets in normed linear spaces

Martini

2023

Banach J. Math. Anal.

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A bounded subset of a (finite or infinite dimensional) normed linear space is said to be complete (or diametrically complete) if it cannot be enlarged without increasing its diameter. Any bounded subset A of a normed linear space is contained in a complete set having the same diameter, which is called a completion of A. We survey characterizations, basic properties, facts about structure of the interior and boundary, and the asymmetry of complete sets. Different methods to obtain completions of bounded sets are presented. Moreover, the structure of the space of complete sets endowed with the Hausdorff metric and relations of this set family to related set families and notions are discussed. For example, we mean here sets of constant width, balls, reduced sets, sets of constant diameter, and sets of constant radius.

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Porosity and diametrical completeness

Cited by 2 publications

References 20 publications

Diametrically Complete Sets with Empty Interior and Constant Width Sets with Empty Interior

Diametrically Complete Sets with Empty Interior and Constant Width Sets with Empty Interior

Complete sets in normed linear spaces

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