Let P:X → V be a projection from a real Banach space X onto a subspace V and let S ⊂ X. In this setting, one can ask if S is left invariant under P, i.e., if PS ⊂ S. If V is finite-dimensional and S is a cone with particular structure, then the occurrence of the imbedding PS ⊂ S can be characterized through a geometric description. This characterization relies heavily on the structure of S, or, more specifically, on the structure of the cone S * dual to S. In this paper, we remove the structural assumptions on S * and characterize the cases where PS ⊂ S. We note that the (so-called) q-monotone shape forms a cone that (lacks structure and thus) serves as an application for our characterization.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 5, pp. 674–684, May, 2012.
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Prophet, M.P., Shevchuk, I.A. Shape-preserving projections in low-dimensional settings and the q-monotone case. Ukr Math J 64, 767–780 (2012). https://doi.org/10.1007/s11253-012-0677-2
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DOI: https://doi.org/10.1007/s11253-012-0677-2