Shape-preserving projections in low-dimensional settings and the q-monotone case

• Main Authors: Prophet, M.P., Shevchuk, I.A.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2012
• Series: Український математичний журнал
• Subjects: Статті
• Online Access: http://dspace.nbuv.gov.ua/handle/123456789/164420

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.