[Math] What are the Necessary and sufficient conditions for concavity and convexity of a two variable function

Question

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 ?

Answer 1

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.