On an elementary inclusion problem and generalized weighted quasi-arithmetic means
Volume 99 / 2013
Banach Center Publications 99 (2013), 45-54
MSC: 39B22.
DOI: 10.4064/bc99-0-4
Abstract
The aim of this note is to characterize the real coefficients $p_1,\dots,p_n$ and $q_1,\dots,q_k$ so that \[\def\conv{\mathop{\rm conv}} \sum_{i=1}^n p_ix_i+\sum_{j=1}^k q_jy_j\in\conv\{x_1,\dots,x_n\} \] be valid whenever the vectors $x_1,\dots,x_n$, $y_1,\dots,y_k$ satisfy \[\def\conv{\mathop{\rm conv}} \{y_1,\dots,y_k\}\subseteq\conv\{x_1,\dots,x_n\}. \] Using this characterization, a class of generalized weighted quasi-arithmetic means is introduced and several open problems are formulated.
