Definition: Bounded Affine Set

Let \(\mathcal A=(A,V_A,v)\) be an \(n\)-dimensional affine space with the associated normed vector space \((V_A,||~||)\) over the field of real numbers \(\mathbb R\)). Let \(P_0,P_1,P_2\ldots,P_s\) be \(s,\, (s\le n)\) affinely independent points. A non-empty subset \(\mathcal U\subset \mathcal B\) of the affine subspace. \[\mathcal B:=\{\lambda_0P_0+\cdots+\lambda_sP_s\,:\,\lambda_i\in \mathbb R, \,i=0,\ldots,s\,\wedge\, \lambda_0+\cdots+\lambda_s=1\}\]

with the designated barycenter. \[B:=\{\lambda(P_0+\cdots+P_s)\,:\,\lambda:=(s+1)^{-1}\in \mathbb R\}\in\mathcal U\]

is called bounded, if there exists an upper bound \(c\in\mathbb R\) with

\[||\overrightarrow{BQ}||\le c\]

for all \(Q\in\mathcal U\).

Definitions: 1

Thank you to the contributors under CC BY-SA 4.0!




  1. Piotrowski, Andreas: Own Research, 2014