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

The Hessian criterion for convexity, in the usual twice-differentiable multivariate setting, isThus 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 $t\mapsto g(x+tv)$, whose second derivative is $v^{\mathsf{T}}H(g)(x+tv)v$. Continuous second partials are a sufficient regularity hypothesis for this line-restriction argument.

Writing $g=e^{x^{\mathsf{T}}x}$, the gradient and Hessian matrix areConsequently, for any nonzero $v$,In particular the Hessian matrix is a positive semidefinite matrix everywhere, so the Hessian criterion for convexity proves $g$ is convex. The strict inequality along every nonconstant line also proves strict convexity.