 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/discrete-mathematics-continuous-mathematics-and-probability/continuous-mathematics/4/b/i/solution

The Hessian criterion for convexity, in the usual twice-differentiable multivariate setting, is

g is convex on R^{d} ⟺ v^{T} H (g) (x) v \geq 0 for all x, v \in R^{d} .

Thus the Hessian matrix must be a positive semidefinite matrix everywhere. The criterion follows by applying the second-derivative criterion for convexity to each line restriction

t \mapsto g (x + t v)

v^{T} H (g) (x + t v) v

. Continuous second partials are a sufficient regularity hypothesis for this line-restriction argument.

 New to topics? Read the docs here!