# Shape-preserving projections in low-dimensional settings and the <i>q </i>-monotone case

### Abstract

Let $P: X \rightarrow V$ be a projection from a real Banach space $X$ onto a subspace $V$ and let $S \subset X$. In this setting, one can ask if $S$ is left invariant under $P$, i.e., if $PS \subset S$. If $V$ is finite-dimensional and $S$ is a cone with particular structure, then the occurrence of the imbedding $PS \subset S$ 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, шє remove the structural assumptions on $S^{*}$ and characterize the cases where $PS \subset S$. We note that the (so-called) $q$-monotone shape forms a cone which (lacks structure and thus) serves as an application for our characterization.
Published

25.05.2012

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 64, no. 5, May 2012, pp. 674-8, http://umj.imath.kiev.ua/index.php/umj/article/view/2607.

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Section

Research articles