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Let $E$ be a set of finite measure, $f$ $\in$ $L^p(E)$, $1<p<∞$. Show that there is a unique $c_*$ such that ${\|f-c_*\|_p}$=$inf_{c\in\mathbb R} \|\ f-c\|_p$.

The existence of $c_0$ can be easily proved by the continuity of the function $F $defined by $F(c)$=${\|f-c\|_p}$. By analyzing the condition of equality in Minkowski's inequality and applying the theory of polynomials I proved the uniqueness when p is an integer. I got stuck when p is not an integer. The problem is that the expansion of ${\|f-c\|_p^p}$ as a power series cannot be easily obtained by Newton's generalized binomial theorem because of the restriction that |x| > |y|. Wikipedia:Binomial theorem Can anyone give me some ideas?

spiritfire
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2 Answers2

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Let assume there exists distinct $c_1$ and $c_2$ such that

$$ \|f-c_1\|_p = \inf_{c\in \mathbb{R}} \|f-c\|_p=\theta \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \mathrm{and} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \|f-c_2\|_p = \inf_{c\in \mathbb{R}} \|f-c\|_p=\theta $$ then by strict convexity of the function $F(c)=\|f-c\|_p$ ( this is a direct consequence of strict convexity of $L^p$ norms for $1<p<\infty$, see Strictly convex Inequality in lp), we have $$\theta \leq \|f-(\frac{c_1+c_2}{2})\|_p < \frac{\|f-c_1\|_p+\|f-c_2\|_p}{2} \leq \theta, $$
a contradiction.

edit: $F(c)=\|f-c\|_p$ is not strict convex but $F(c)^p=\|f-c\|^p_p$ is strict convex and you can adapt the proof.

edit: The strict inequality holds if $f$ is not constant.

Community
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Fin8ish
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  • Your map isn't strictly convex. For example, if $f \equiv 0$, then the map is just $|c|$ and merely convex. However, $|f - c|_p^p$ is strictly convex. – Dominik Dec 25 '16 at 10:34
  • @Dominik : sry. yes you are right. Now it is enough to change $|f-c|_p$ with $|f-c|^p_p$, the proof still works. – Fin8ish Dec 25 '16 at 11:07
  • Thank you for your hyperlink. I didn't simplify enough the necessary and sufficient condition for equality in Minkowski's inequality. I edited your answer according to Dominik. – spiritfire Dec 25 '16 at 11:42
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Hint: Show that the map $$c \mapsto \|f - c\|_p^p$$ is strictly convex for $p > 1$.

Dominik
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