Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $f\in C([0,1])$ and $K\subset C([0,1])$ be the set of constant functions on $[0,1]$. Let $\|u\|=\sup\{|u(x)|:\ x\in [0,1]\}$. Define $F:C([0,1])\rightarrow \mathbb{R}$ by $$F(g)=\|f-g\|$$

Consider the problem of minimize $F$ in $K$ and let $c\in K$ be the minimum.

1 - Is it possible to characterize $c$ in terms of $f$.

2 - Is it possible to characterize $F(c)$.

share|cite|improve this question
The solution isn't simply $$c=\frac{\sup\{f\}+\inf\{f\}}{2}\ ?$$ Am I missing something? – yohBS Dec 20 '12 at 14:16
up vote 1 down vote accepted

Hint: Yes, it is possible. Just think of it, the difference between a constant $c$ and $f$ is given by $$ \|f - c\| = \max\{ |c - \max f|, |c - \min f|\} $$ Why? Now for which $c$ is this minimal? Think of balancing both terms ...

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.