Polyhedral points, QP-points and Q-points. (English) Zbl 07812640
In the present paper the relation between polyhedral points, \(QP\)-points and \(Q\)-points of the unit sphere of a Banach space is studied.
A Banach space is said to be polyhedral if the closed unit ball of every finite dimensional subspace is a polygon. According to [R. Durier and P. L. Papini, Rocky Mt. J. Math. 23, No. 3, 863–875 (1993; Zbl 0799.46015)], a Banach space \(X\) is polyhedral if and only if each point of \(B_X\) is a polyhedral point. A Banach space is a \(QP\)-space (\(Q\)-space, resp.) if each point of its unit sphere is a \(QP\)-point (\(Q\)-point, resp.). In [D. Amir and F. Deutsch, J. Approx. Theory 6, 176–201 (1972; Zbl 0238.41014)], it was proved that a \(QP\)-space is a Polyhedral space.
In Theorem 2.7, the authors prove the local version of the previous result: for a point \(x\) in the unit sphere of a Banach space, it holds that \[ x \mbox{ is a QP-point}\Rightarrow x \mbox{ is a polyhedral point}\Rightarrow x \mbox{ is a Q-point}. \] In Section 2, it is also proved that the last implication can be reversed if in addition the point \(x\) is a weakly polyhedral point of the unit sphere.
In Section 3 it is proved the local version of some results contained in [L. Veselý, Extr. Math. 15, No. 1, 213–217 (2000; Zbl 0987.46019)]. In particular, it is proved that if \(x\) is a polyhedral point of the unit sphere of a Banach space, \(x\) is a Gâteaux smooth point if and only if the relative interior of a maximal face of \(B_X\) contains \(x\).
Finally, Section 4 contains some geometrical properties of polyhedral points.
A Banach space is said to be polyhedral if the closed unit ball of every finite dimensional subspace is a polygon. According to [R. Durier and P. L. Papini, Rocky Mt. J. Math. 23, No. 3, 863–875 (1993; Zbl 0799.46015)], a Banach space \(X\) is polyhedral if and only if each point of \(B_X\) is a polyhedral point. A Banach space is a \(QP\)-space (\(Q\)-space, resp.) if each point of its unit sphere is a \(QP\)-point (\(Q\)-point, resp.). In [D. Amir and F. Deutsch, J. Approx. Theory 6, 176–201 (1972; Zbl 0238.41014)], it was proved that a \(QP\)-space is a Polyhedral space.
In Theorem 2.7, the authors prove the local version of the previous result: for a point \(x\) in the unit sphere of a Banach space, it holds that \[ x \mbox{ is a QP-point}\Rightarrow x \mbox{ is a polyhedral point}\Rightarrow x \mbox{ is a Q-point}. \] In Section 2, it is also proved that the last implication can be reversed if in addition the point \(x\) is a weakly polyhedral point of the unit sphere.
In Section 3 it is proved the local version of some results contained in [L. Veselý, Extr. Math. 15, No. 1, 213–217 (2000; Zbl 0987.46019)]. In particular, it is proved that if \(x\) is a polyhedral point of the unit sphere of a Banach space, \(x\) is a Gâteaux smooth point if and only if the relative interior of a maximal face of \(B_X\) contains \(x\).
Finally, Section 4 contains some geometrical properties of polyhedral points.
Reviewer: Jacopo Somaglia (Milano)
MSC:
| 46B20 | Geometry and structure of normed linear spaces |
| 52A01 | Axiomatic and generalized convexity |
| 52A07 | Convex sets in topological vector spaces (aspects of convex geometry) |
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