According to https://en.wikipedia.org/wiki/Positive-definite_matrix: "It turns out that the (hessian) matrix M (of a multi-dimensional function) is positive definite if and only if it is symmetric and its quadratic form is a strictly convex function." i.e. Convexity seems to imply positive (semi-)definiteness.
Is there an intuitive (possibly geometric) explanation for why this is the case?
I know that the diagonals of the hessian matrix of a function give the curvature of that function along the respective dimensions. For convexity to hold, the multidimensional function must have a positive curvature in every dimension (all diagonals >= 0) and in every possible combination of those dimensions. How does positive (semi-)definiteness ensure this?