convex set ConvexSet 2001-10-15 23:32:17 2007-06-18 20:06:43 Definition polyconvex set polyconvex \usepackage{graphicx} %\usepackage{xypic} \usepackage{bbm} \newcommand{\Z}{\mathbbmss{Z}} \newcommand{\C}{\mathbbmss{C}} \newcommand{\R}{\mathbbmss{R}} \newcommand{\Q}{\mathbbmss{Q}} \newcommand{\mathbb}[1]{\mathbbmss{#1}} \newcommand{\figura}[1]{\begin{center}\includegraphics{#1}\end{center}} \newcommand{\figuraex}[2]{\begin{center}\includegraphics[#2]{#1}\end{center}} Let $S$ a subset of $\mathbb{R}^n$. We say that $S$ is \emph{convex} when, for any pair of points $A,B$ in $S$, the segment $\overline{AB}$ lies entirely inside $S$.\smallskip The former statement is equivalent to saying that for any pair of vectors $u,v$ in $S$, the vector $(1-t)u+tv$ is in $S$ for all $t\in[0,1]$.\smallskip If $S$ is a convex set, for any $u_1,u_2,\ldots,u_r$ in $S$, and any positive numbers $\lambda_1,\lambda_2,\ldots,\lambda_r$ such that $\lambda_1+\lambda_2+\cdots+\lambda_r=1$ the vector $$\sum_{k=1}^r\lambda_k u_k$$ is in $S$.\medskip Examples of convex sets in the plane are circles, triangles, and ellipses. The definition given above can be generalized to any real vector space: Let $V$ be a vector space (over $\R$ or $\C$). A subset $S$ of $V$ is \emph{convex} if for all points $x,y$ in $S$, the line segment $\{\alpha x + (1-\alpha) y \mid \alpha\in(0,1)\} $ is also in $S$. More generally, the same definition works for any vector space over an ordered field. A \emph{polyconvex set} is a finite union of compact, convex sets.