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Consider any real-valued convex function [itex]c : R^n \rightarrow R[/itex]. I am interested in whether there exists some strictly convex function d, that satisfies d(x) > d(y) if c(x) > c(y).

That is, given a convex function, can we always find a strictly convex function that preserves strict inequalities between points?

Originally I thought this would be straightforward, but now I am not so sure.

If we restrict ourselves to considering a function [itex]c : R \rightarrow R[/itex] then one way we can think of attempting a solution might be to take the minima, or set of minima, and choose one arbitrarily, then divide the the function with a vertical plane and add some increasing with a strictly increasing derivative function to either side of the division. This would create a strictly convex function but not preserve the ordering. The ordering is preserved to either side of the division, but the relative ordering of both sides is now unclear.

Of course, the question does not call for the construction of such a strictly convex function, but I can see no immediate steps to a nonconstructive proof, either.

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# Existence of strictly convex functions with same ordering as convex one

Can you offer guidance or do you also need help?

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