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jrand
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Homework Statement
I am trying to understand the reasoning behind the following statement from Boyd's Convex Optimization textbook, page 50, line 9-11: "f must be negative on C; for if f were zero at a point of C then f would take on positive values near the point, which is a contradiction." The assumptions are: f is an affine function, C is an open convex set.
Homework Equations
The Attempt at a Solution
I know: f is an affine function of the form [tex]f(x) = a^{T}x+b[/tex]. The image of a convex set S under f is convex. That is, if I take all the elements of S, and input them into f, the result is another convex set.
I also know: C is an open convex set. That is, for any two points in C, the line segment between them is contained in C. The most important point of the problem is C is open, which means whatever shape C is, the boundary or border is not contained in C. Somehow this fact (open convex set) is the reason behind the initial quoted statement but I do not understand why.
Thanks in advance for any help or tips.