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For concavity and convexity is that the Hessian be negative semidefinite and positive semidefinite But for strict concavity definiteness is only the sufficient condition not the necessary . What is the necessary condition for a two variable function to ensure strict concavity / convexity ?

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Your initial statement is wrong: the Hessian does not have to exist for the function to be concave or convex. For example, $|x|$ is convex but does not have derivatives at $0$.

A necessary and sufficient condition for a convex/concave function to be strictly convex/concave is that its graph does not contain any line segment.

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  • $\begingroup$ so the only way to do that is by checking it graphically? No other way? $\endgroup$
    – Riya Arora
    Commented Sep 13, 2016 at 5:49
  • $\begingroup$ You asked for a necessary and sufficient condition, I gave one. How you check it is another matter: it may depend on how the function is defined. $\endgroup$ Commented Sep 13, 2016 at 15:03

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