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Plasticity of the unit ball of a strictly convex Banach space. (English) Zbl 1362.46011

If \((X,d)\) is a metric space, a mapping \(f:X\to X\) is noncontractive if \(d(f(x), f(y)) \geq d(x,y)\) for every \(x, y\in X\). The metric space \(X\) is an expand-contract plastic space (an EC-space) if every noncontractive bijection from \(X\) onto itself is an isometry, or, equivalently, if every nonexpansive bijection from \(X\) onto itself is an isometry. The authors prove that the closed unit ball of a strictly convex Banach space is an EC-space and give an example of a closed bounded convex set in a Hilbert space that is not an EC-space. Whether the closed unit ball of every Banach space is an EC-space is an open question.

MSC:

46B20 Geometry and structure of normed linear spaces
54E40 Special maps on metric spaces
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References:

[1] Freudenthal, H., Hurewicz, W.: Dehnungen, Verkürzungen, isometrien. Fund. Math. 26, 120-122 (1936) · JFM 62.0690.03
[2] Mankiewicz, P.: On extension of isometries in normed linear spaces. Bull. Acad. Polon. Sci., Sér. Sci. Math. Astronom. Phys. 20, 367-371 (1972) · Zbl 0234.46019
[3] Naimpally, S.A., Piotrowski, Z., Wingler, E.J.: Plasticity in metric spaces. J. Math. Anal. Appl. 313, 38-48 (2006) · Zbl 1083.54016
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