---
title: "convexity preserve line restriction"
source: https://www.jemoka.com/posts/kbhconvexity_preserve_line_restriction/
---

\(f: \mathbb{R}^{n} \to \mathbb{R}\) is convex IFF the function \(g: \mathbb{R} \to \mathbb{R}\) is convex:
\begin{equation} g\qty(t) = f\qty(x + tv), \text{dom } g = \qty {t \mid x + tv \in \text{dom }f} \end{equation}
is convex in \(t \in \mathbb{R}\) for any \(x \in \text{dom } f\), \(v \in \mathbb{R}^{n}\).
