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@jemoka / Jemoka Knowledge Base / raw/concept/kbhconvexity_preserve_line_restriction.md
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--- 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 \(x \in \text{dom } f\), \(v \in \mathbb{R}^{n}\).