@article{Prophet_Shevchuk_2012, title={Shape-preserving projections in low-dimensional settings and the <i>q </i>-monotone case}, volume={64}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/2607}, abstractNote={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.}, number={5}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={Prophet, M. P. and Shevchuk, I. A.}, year={2012}, month={May}, pages={674-684} }