# Strongly convex function

Arises for $LaTeX: \textstyle f \in C^2:$ eigenvalues of hessian are bounded away from zero (from below): there exists $LaTeX: \textstyle K > 0$ such that $LaTeX: \textstyle h' H_f(x) h \ge K||h||^2$ for all $LaTeX: \textstyle h \in \mathbb{R}^n.$ For example, the function $LaTeX: \exp(-x)$ is strictly convex on $LaTeX: \mathbb{R},$ but its second derivative is $LaTeX: \textstyle \exp(-x),$ which is not bounded away from zero. The minimum is not achieved because the function approaches its infimum of zero without achieving it for any (finite) $LaTeX: x.$ Strong convexity rules out such asymptotes.