---
title: "function convexity conditions"
type: concept
source: https://www.jemoka.com/posts/kbhfunction_convexity_conditions/
confidence: high
status: active
---

1st order condition differentiable \(f\) with convex domain is convex IFF:
\begin{equation} f\qty(y) \geq f\qty(x) + \nabla f\qty(x)^{T} \qty(y-x), \forall x,y \in \text{dom } f \end{equation}
&ldquo;the function is everywhere above the Taylor approximation&rdquo; =&gt; &ldquo;first-order Taylor approximation of \(f\) is a global underestimator of \(f\).&rdquo;
2nd order condition for twice differentiable \(f\) with convex domain, we have:
\(f\) is convex IFF \(\nabla^{2} f\qty(x) \succeq 0, \forall x \in \text{dom } f\) (i.e. that the Hessian is PSD) if \(\nabla^{2} f\qty(x) \succ 0 \forall x \in \text{dom } f\), then we call \(f\) strictly convex (i.e. that the Hessian is PD) you may enjoy using Cauchy-Schwartz Inequality to show these.
We also call the PSD-ness of \(\nabla^{2}f\qty(x)\) the &ldquo;curvature.&rdquo;
