document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=25800>Question 25800</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove that R(to the nth)is a convex set......\r<BR>\n" );
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document.write( "can I just use 2 points X1 and X2 but how do I prove from then?\r<BR>\n" );
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document.write( "p.s. not sure if this is the correct area to post.... </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #13885 by </B><A HREF=/tutors/aboutme.mpl?userid=kev82><B>kev82(151)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=kev82><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=13885\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Hi,\r<BR>\n" );
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document.write( "I don't know what definition of convex you're working with, but they shouldn't differ too much, so here's mine.\r<BR>\n" );
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document.write( "A set is convex, if for all pairs of elements of the set, the geodesic lies entirely within the set.\r<BR>\n" );
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document.write( "The geodesic is the shortest path, in <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\mathbb{R}^n\"> it is always a straight line.\r<BR>\n" );
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document.write( "So lets pick two arbitrary points <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?x_1\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?x_2\">, and find the straight line between them.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?l(t)=x_1+t(x_2-x_1)\;t\in\[0,1\]\">\r<BR>\n" );
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document.write( "Because <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\mathbb{R}^n\"> is a vector space, we can add and scalar multiply to our heart's content, without leaving <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\mathbb{R}^n\">. Thus <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?l(t)\in\mathbb{R}^n\,\forall t\">\r<BR>\n" );
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document.write( "And that's it, proved.\r<BR>\n" );
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document.write( "Hope that helps, if you have any problems, please email back.\r<BR>\n" );
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document.write( "Kev<BR>\n" );
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document.write( "Question from original poster<BR>\n" );
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document.write( "Thanks......I have my line segment as L=mX1+(1-m)X2     0<=m<=1<BR>\n" );
document.write( "is this any different then the line that you had?<BR>\n" );
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document.write( "There are two ways to think about this. You can say that any two straight lines that go through the same two points must be the same.\r<BR>\n" );
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document.write( "My line passes through <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?x_1\"> when <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?t=0\"> and it passes through <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?x_2\"> when <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?t=1\">. Your line passes through <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?x_1\"> when <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?m=1\"> and passes through <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?x_2\"> when <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?m=0\">. So we have two straight lines that each pass through two points, therefore the must be the same.\r<BR>\n" );
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document.write( "The other much better(If you like geometry, which I do) way to think about it is this.\r<BR>\n" );
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document.write( "Think of a person travelling along my line, now for any <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?t\in\[0,1\]\"> they are on the line. Lets consider a general function s(x), that maps the closed unit interval to its self. <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?s:\[0,1]\to\[0,1]\">. Because <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?s(x)\in\[0,1\]\"> then choosing <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?t=s(x)\">, we know that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?l(s(x))\"> is still the same line. Although the person moves along it differently. If we can choose an <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?s(m)\"> that makes <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?l(s(m))=L(m)\"> then the two lines are clearly the same(note that this technique works in general for any path, not just lines)\r<BR>\n" );
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document.write( "Let's choose <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?s(m)=1-m\">, so <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?l(s(m))=x_1+(1-m)(x_2-x_1)\"> If you tidy this up you will find you get the same equation, thus the lines are the same.\r<BR>\n" );
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document.write( "Hope that helps,<BR>\n" );
document.write( "Kev. </SPAN>\n" );
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