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Math Help - convexity in euclidean vector space

  1. #1
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    convexity in euclidean vector space

    If S is a convex polyhedron in a euclidean vector space U, and f:S->R is a convex function, then does f necessarily attain a maximum on S? and how would i prove it?
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    Quote Originally Posted by squarerootof2 View Post
    If S is a convex polyhedron in a euclidean vector space U, and f:S->R is a convex function, then does f necessarily attain a maximum on S? and how would i prove it?
    Consider any line segment  \lambda \bold{x}+(1-\lambda)\bold{y}, \lambda \in [0,1] which passes through the maximum of f ,where \bold{x} and \bold{y} on S.

    Because the polyhedron is convex the line segment has its end points on S and is otherwise inside or on the poyhedron. But because f is convex we have:

     <br />
f(\lambda \bold{x}+(1-\lambda)\bold{y})\le \lambda f(\bold{x})+(1-\lambda)f(\bold{y}) \le \max(f(\bold{x}),f(\bold{y}))<br />

    Which proves that one of the end points is the maximum.

    RonL
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    i was thinking about this problem, and what about the theorem that says if C is a convex set and f is a convex function on C, and if f attains its maximum on C at an interior point of C, then f is constant on C ?(proof can be found here on pg. 194: Topological Spaces: Including a ... - Google Book Search)

    then wouldnt f attaining its maximum at an interior point of the set imply that there is no maximum? (since f is constant)
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    Quote Originally Posted by squarerootof2 View Post
    i was thinking about this problem, and what about the theorem that says if C is a convex set and f is a convex function on C, and if f attains its maximum on C at an interior point of C, then f is constant on C ?(proof can be found here on pg. 194: Topological Spaces: Including a ... - Google Book Search)

    then wouldnt f attaining its maximum at an interior point of the set imply that there is no maximum? (since f is constant)
    If f is a constant then every point is a maximum including the boundary, so no problem there.

    In fact in my earlier post I did not clain (nor wish to imply) that the maximum at an endpoit of the line segment was unique.

    We have the situation that you mention of the function being a constant, it is also posible if the function is linear for all the points on a face or edge etc to be maxima.

    RonL
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    what if we consider in R^2 the convex function f(x,y)=x on the set S={f(x,y) such that y>=0}? then it does not attain its maximum on S even though y>=0 is a convex polyhedron and f is a convex function. wouldn't this be a counterexample to what we are trying to prove?
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    Quote Originally Posted by squarerootof2 View Post
    what if we consider in R^2 the convex function f(x,y)=x on the set S={f(x,y) such that y>=0}? then it does not attain its maximum on S even though y>=0 is a convex polyhedron and f is a convex function. wouldn't this be a counterexample to what we are trying to prove?
    You are using the term polyhedron, and I am assuming that you mean a convex polytope, which is closed, your set S here is not a convex polytope.

    RonL
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    ok sorry for not stating the definition of polyhedron that i am using. basically the definition im working with is "the intersection of finitely many closed halfspaces is called a polyhedron." a closed and bounded polyhedron is called a polytope.

    so under this definition, would my counterexample suffice? (i.e. would the restrictions given in the problem permit my counterexample to work?)
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    Quote Originally Posted by squarerootof2 View Post
    ok sorry for not stating the definition of polyhedron that i am using. basically the definition im working with is "the intersection of finitely many closed halfspaces is called a polyhedron." a closed and bounded polyhedron is called a polytope.

    so under this definition, would my counterexample suffice? (i.e. would the restrictions given in the problem permit my counterexample to work?)
    Sorry but I'm now lost, the last thing that I see is that f attained its maximum at an interior point, but now we have to maximum, the problem presupposed that there was a maximum.

    RonL
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    this is a different example. i'm asking if we have such a function that f(x,y)=x, and S is the set of all the values f(x,y) can take such that y is greater than or equal to zero, then do we have a convex polyhedron with a convex function mapping from S to R such that the maximum is not attained on S? (under the definition of polyhedron i gave)
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