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Geometry of isometric reflection vectors. (English) Zbl 1274.46035

It is proved that, if \(X\) is a Banach space and \(F\) is a minimal face of the unit ball in \(X\) containing an isometric reflection vector, then \(F\) has to be an exposed face (Theorem 2.1). Also, in Theorem 3.1 there is given an example of a Banach space containing a norm one vector \(e\) being an isometric reflection vector in every two dimensional subspace containing it, which is not an isometric reflection vector in the whole space. Recall that a norm one vector \(e\) in a Banach space \(X\) is said to be an isometric reflection vector if there is a closed and maximal subspace \(M\) of \(X\) such that \(X=M\oplus \langle e \rangle \) and \(\| te+m\| =\| te-m\| \) for every \(m\in M\) and \(t\in \mathbb R\).

MSC:

46B20 Geometry and structure of normed linear spaces
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