The questions can also be put like the following. I believe these three questions make it easier for you to answer exactly.

(1) Given a twice continuously differentiable function , it can be justified that is not always positive for . However, if , is ("locally") convex in some epsilon distance around ? (As shown in the 1st picutre in #1)

(2) Given a twice continously differentiable function , it can be justified that Hessian Matrix of is not always postive definite for . However, if Hessian of at is positve definite, is ("locally") convex in some epsilon neighborhood of ?

(3) Given a region defined by and and (usually defines the feasible region of a general constrained optimization problem), where every and is twice continously differentiable. Here is not convex for , is not affinely linear, so is not a convex set "as a whole". But for a feasible point , on what condition (I would like to know condition about and ,not just the "at + (1-t)b" definition of convexity set), a neighborhood of in is ("locally") convex? (As shown in the 2nd picture in #1)

As to this question, if this kind of condition exists, Hessian of and is probably involved, as I guessed.

Thanks very much!