TY - JOUR
AU - M. P. Prophet
AU - I. A. Shevchuk
PY - 2012/05/25
Y2 - 2023/03/31
TI - Shape-preserving projections in low-dimensional settings and the <i>q </i>-monotone case
JF - Ukrains’kyi Matematychnyi Zhurnal
JA - Ukr. Mat. Zhurn.
VL - 64
IS - 5
SE - Research articles
DO -
UR - https://umj.imath.kiev.ua/index.php/umj/article/view/2607
AB - 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.
ER -