Strictly convex function, implies that translation is strictly convex.

Question

I'm wondering about this question. Prove that the function $f(x)=\|x\|^{2}$, and then answer. If the function $f(x)=\|x\|^{2}$ is strictly convex implies that $f(x)=\|x-a\|^{2}$ is strictly convex? Where $a\in \mathbb{R}^{n}$. I'm available to prove that $\|x\|^{2}$, but i'm not sure about of this question.

Can someone give me a light? Thanks.

If you have proved that $| \cdot |$ is strictly convex then the translation follows immediately from the defintion. — copper.hat, Sep 22 '22 at 16:34

score 1 · Answer 1 · answered Sep 22 '22 at 16:26

1

If we have a strict convex function $f$, you can prove strict convexity of translation by doing something like \begin{align*} &f(tx + (1-t)y - a) \\ &= f(tx + (1-t)y - ta - (1-t)a) \\ &= f(t(x-a) + (1-t)(y-a)) \\ &< tf(x-a) + (1-t)f(y-a) \\ \end{align*}

answered Sep 22 '22 at 16:26

gist076923

1,266

Thanks! @gst076923. I'm not sure if the two first lines in the proof are necessary. Thanks! again. – Skinner. Sep 22 '22 at 20:36

Strictly convex function, implies that translation is strictly convex.

1 Answers1