Other than convenience in convex optimization, is there a reason that the definition of a convex function includes the requirement for the domain to be a convex set?
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3$\begingroup$ How would you define a convex function on a nonconvex domain? $\endgroup$– Michael GreineckerCommented Jan 10, 2017 at 16:41
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$\begingroup$ Alternatively, if one takes the epigraph definition ($f$ is convex is its epigraph is a convex set) then the effective domain of $f$ is necessarily convex. See this and other discussions on page 23 of Rockafeller. $\endgroup$– Willie WongCommented Jan 10, 2017 at 16:52
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$\begingroup$ I would imagine that the definition "I every point there is hyperplane, touching the graph in that point and lying below the graph globally" would work… I don't see any benefit of this, though. $\endgroup$– DirkCommented Jan 10, 2017 at 20:10
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1$\begingroup$ In arbitrary domain one can define the notion of locally convex function, i.e. function which is convex in a small convex neighborhood of any point. In case of convex domain this notion of locally convex function is equivalent to the usual notion of convex function. $\endgroup$– asvCommented Jan 11, 2017 at 6:18
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$\begingroup$ Sorry, this was supposed to go on Math StackExchange. My confusion is like this: if convexity means that I have to be able to draw a line segment between any two points in a graph, and all f(x) within the interval defined by said line have to be below that line, this would hold even if the domain was not convex (e.g. multiple intervals on the reals). $\endgroup$– user32849Commented Jan 12, 2017 at 10:29
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