# Thread: convexity in euclidean vector space

1. ## 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?

2. Originally Posted by squarerootof2
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 $\displaystyle \lambda \bold{x}+(1-\lambda)\bold{y}$, $\displaystyle \lambda \in [0,1]$ which passes through the maximum of $\displaystyle f$,where $\displaystyle \bold{x}$ and $\displaystyle \bold{y}$ on $\displaystyle S$.

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

$\displaystyle f(\lambda \bold{x}+(1-\lambda)\bold{y})\le \lambda f(\bold{x})+(1-\lambda)f(\bold{y}) \le \max(f(\bold{x}),f(\bold{y}))$

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

RonL

3. 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)

4. Originally Posted by squarerootof2
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

5. 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?

6. Originally Posted by squarerootof2
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

7. 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?)

8. Originally Posted by squarerootof2
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

9. 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)