convex hull of finite set is compact - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-26T07:40:08Z http://mathoverflow.net/feeds/question/22562 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22562/convex-hull-of-finite-set-is-compact convex hull of finite set is compact santhosh 2010-04-26T05:42:51Z 2011-11-04T11:01:45Z <p>In a Banach space, is the convex hull of finite set compact?</p> http://mathoverflow.net/questions/22562/convex-hull-of-finite-set-is-compact/22577#22577 Answer by Matthew Daws for convex hull of finite set is compact Matthew Daws 2010-04-26T09:52:56Z 2010-04-26T09:52:56Z <p>Actually, the convex hull of a sequence of points $(x_n)$ is (relatively) compact when $x_n\rightarrow 0$, and this easily gives a positive answer to your question (but is somewhat overkill). In fact, a closed convex set K in a Banach space is compact if and only if it's contained in the closed convex hull of a sequence $(x_n)$ with $x_n\rightarrow 0$. See, for example, Lindenstrauss and Tazfriri, vol I, Proposition 1.e.2.</p> http://mathoverflow.net/questions/22562/convex-hull-of-finite-set-is-compact/22578#22578 Answer by Qfwfq for convex hull of finite set is compact Qfwfq 2010-04-26T10:15:18Z 2010-04-26T10:15:18Z <p>Of course yes: the n points lie in the finite-dimentional linear subspace generated by themselves (remember that any norm, when restricted to a finite-dimensional linear subspace, gives rise to the same topology on that space).</p> http://mathoverflow.net/questions/22562/convex-hull-of-finite-set-is-compact/22608#22608 Answer by Mariano Suárez-Alvarez for convex hull of finite set is compact Mariano Suárez-Alvarez 2010-04-26T15:20:50Z 2010-04-26T15:20:50Z <p>Suppose $X$ is your Banach space and let $\{x_1,\dots,x_n\}$ be a finite subset of $X$. Let $$S=\{(t_1,\dots,t_n)\in\mathbb R^n:t_1,\dots,t_n\geq0,\,t_1+\cdots+t_n=1\}$$ be the <em>standard simplex</em> in $\mathbb R^n$. The map $$\phi:(t_1,\dots,t_n)\in S\mapsto t_1x_1+\cdots+t_nx_n\in X$$ is evindently continuous and its image is $\mathrm{conv}\{x_1,\dots,x_n\}$. Since $S$ is compact, so is $\mathrm{conv}\{x_1,\dots,x_n\}$.</p> http://mathoverflow.net/questions/22562/convex-hull-of-finite-set-is-compact/31524#31524 Answer by Oleg Reinov for convex hull of finite set is compact Oleg Reinov 2010-07-12T10:29:55Z 2010-07-12T10:29:55Z <p>Just apply Induction for finite dim spaces with Dim (n points).</p> http://mathoverflow.net/questions/22562/convex-hull-of-finite-set-is-compact/80035#80035 Answer by Oleg Reinov for convex hull of finite set is compact Oleg Reinov 2011-11-04T11:01:45Z 2011-11-04T11:01:45Z <p>Sorry, but I did not answer the question as well as another one, ever. I do not understand why this "Just apply Induction for finite dim spaces with Dim (n points)." appeared as my answer... Oleg Reinov</p>