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I'm working on a problem in game theory, and I've run into a proposition that I don't know how to prove. Leaving out the game theory bit, I have a family of functions defined over a k-dimensional simplex onto the real numbers. Each function is fairly well-behaved: continuous and differentiable in most places, possibly including a small number of discontinuities.

Based on simulations, I strongly suspect that with measure one, a function within this class has a unique global maxima. How do I prove it?

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    I can try to verify if they are concave for example - or maybe convex. Please provide this family of functions, it can ease the help to you.2011-06-22
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    Unfortunately, the functions aren't convex or concave -- that would be too easy :). If it's helpful, I can give more detail on the class of functions. But it seems like "one global max" should be a property of a lot of functions. To put it another way, the set of functions with two exactly equal global maxima should be infinitesimally small, right...? I was hoping there'd be a general proof for this.2011-06-22
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    If you can give more detail on the class of functions (e.g. an explicit parameterization, or a set of criteria which all functions in the class satisfy) then we will be able to give you a better answer.2011-06-22

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