A new characterization of strict convexity on normed linear spaces

Nermin Okicic, Amra Rekic-Vukovic, Vedad Pasic

Abstract


We consider relations between the distance of a set \(A\) and the distance of its translated set \(A+x\) from 0, for \(x\in A\), in a normed linear space. If the relation \(d(0,A+x)<d(0,A)+\|x\|\) holds for exactly determined vectors \(x\in A\), where \(A\) is a convex, closed set with positive distance from 0, which we call (TP) property, then this property is equivalent to strict convexity of the space. We show that in uniformly convex spaces the considered property holds.

Keywords


Translation; uniform convexity; strict convexity

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References


Ayerbe Toledano, J. M., Domınguez Benavides, T., López Acedo, G., Measures of Noncompactness in Metric Fixed Point Theory, Birkhauser Verlag, Basel, 1997.

Cobzas, S., Geometric properties of Banach spaces and the existence of nearest and farthest points, Abstr. Appl. Anal. 2005(3) (2005), 259–285.

Istratescu, V. I., Strict Convexity and Complex Strict Convexity: Theory and Applications, Marcel Dekker, Inc., New York, 1984.




DOI: http://dx.doi.org/10.17951/a.2022.76.1.61-71
Date of publication: 2022-10-05 20:39:35
Date of submission: 2022-10-04 21:20:10


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