Results 1 to 6 of 6
Like Tree6Thanks
  • 2 Post By Plato
  • 2 Post By HallsofIvy
  • 2 Post By Plato

Thread: convex setssetst

  1. #1
    No one in Particular VonNemo19's Avatar
    Joined
    Apr 2009
    From
    Detroit, MI
    Posts
    1,849

    convex set

    I'm self studying analysis and I'm going to need help from time to time conceptualizing the ideas. First off...Can someone touch on how they think about convex sets? If you could explain why all the points on a line between x and y is given by \lambda{x}+(1-\lambda)y that would be helpful.
    Last edited by VonNemo19; Feb 18th 2017 at 02:33 PM. Reason: ii
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,101
    Thanks
    2572
    Awards
    1

    Re: convex set

    Quote Originally Posted by VonNemo19 View Post
    I'm self studying analysis and I'm going to need help from time to time conceptualizing the ideas. First off...Can someone touch on how they think about convex sets?
    You must be more exact in telling us exactly in what domain of definition you are working.
    Are you working in a Euclidean metric, in a plane, is 3-space or general space?

    Why that matters is clear from the description of convexity: in a convex region if $P~\&~Q$ are two points in the region and point $R$ is between them the point $R$ is also in the region. That is abstract in that it depends heavily upon the definition of betweenness.

    In an ordinary Euclidean plane a set is convex if $P~\&~Q$ are two points in the set then the line segment $\overline{PQ}$ is a subset of the set. It is easy to see that a triangle with its interior is convex. The same for a circle, parallelogram, trapezoid. etc.

    Think of a star shaped figure with its interior, it is easy to see that it is not convex. Pick two points in different peaks of the star. The line segment connecting them does not 'remain in' the star.
    Thanks from topsquark and VonNemo19
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,031
    Thanks
    2766

    Re: convex setssetst

    Given any two points in a convex set, every point on the line segment between them is in the set.

    It is not true that all the points on a line between x and y are given by [tex]\lambda x+ (1- \lambda) y).

    It is true that every point given by that is on the line passing through x and y but they are between x and y only for 0< \lambda< 1. That it true because, first, this is linear in \lambda so describes a straight line, second, when \lambda= 0 we get y, and, third, because when \lambda= 1 we get x. If \lambda< 0 we get points on the line but not "between" 0 and 1, they are on the side of y away from x. And if \lambda> 1 we get points on the line but on the side of x away from y.
    Thanks from topsquark and VonNemo19
    Follow Math Help Forum on Facebook and Google+

  4. #4
    No one in Particular VonNemo19's Avatar
    Joined
    Apr 2009
    From
    Detroit, MI
    Posts
    1,849

    Re: convex setssets

    Thanks. You guys have helped enormously just talking about it with me for a second. The definition I'm working with is as follows:

    We call a set E\subset{R^k} convex if \lambda\bold{x}+(1-\lambda)\bold{y}\in{E}

    whenever \bold{x}\in{E} , \bold{y}\in{E}, and 0<\lambda<1

    I got the line idea from some website after I googled convex sets. I'm just trying to visualize what's happening.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,101
    Thanks
    2572
    Awards
    1

    Re: convex setssets

    Quote Originally Posted by VonNemo19 View Post
    We call a set E\subset{R^k} convex if \lambda\bold{x}+(1-\lambda)\bold{y}\in{E} whenever \bold{x}\in{E} , \bold{y}\in{E}, and 0<\lambda<1
    I'm just trying to visualize what's happening.
    Allow me to be simply honest. You will never be completely comfortable with this material until you are comfortable working in general vector spaces. That is the point that Prof. HallsofIvey is making. $\bf{x}+\lambda(\bf{y}-\bf{x})$ where $ \bf{x}~\&~\bf{y}$ are vectors and $\lambda$ is a scalar is a line in the space. But if $0\le\lambda\le 1$ is a line segment between $\bf{x}~\&~\bf{y}$. Please note the $\le$ is incorrect in your definition.

    Try to visualize the difference in a line and a line segment. One is bounded and one is not bounded.
    One is essential in the definition of convexity, the other does not.
    Thanks from HallsofIvy and VonNemo19
    Follow Math Help Forum on Facebook and Google+

  6. #6
    No one in Particular VonNemo19's Avatar
    Joined
    Apr 2009
    From
    Detroit, MI
    Posts
    1,849

    Re: convex setssets setssets

    Quote Originally Posted by Plato View Post
    Allow me to be simply honest. You will never be completely comfortable with this material until you are comfortable working in general vector spaces. .
    Honesty is what I'm after. Sugarcoating my inadequacy in certain areas will do nothing for me but slow me down. I need your replies to be matter of fact so that the space I work in is convex, allowing me to move in straight lines from points of ignorance to those of understanding. By the way, I skipped linear algebra because I found it dull. If I come up against something I can't get because of this, I'll take the time necessary to go back and learn it. I've done this with vectors already so that my understanding of convex sets is adequate for now.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. convex function and convex set
    Posted in the Calculus Forum
    Replies: 8
    Last Post: Jul 10th 2013, 01:09 AM
  2. Convex hull of the union of two convex sets
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: May 24th 2013, 01:01 PM
  3. The union of two convex sets is not convex
    Posted in the Discrete Math Forum
    Replies: 4
    Last Post: Jan 30th 2010, 03:23 PM
  4. Proving that max{0,f(x)} is convex if f is convex
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Nov 5th 2009, 06:16 AM
  5. Proving that f^2 is convex if f is convex and f>=0
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Nov 3rd 2009, 09:51 AM

/mathhelpforum @mathhelpforum