Convex Set in R^n Problem

[Onezimo Cardoso]

Homework Statement

Let $C \subset \mathbb{R}^n$ a convex set. If $x \in \mathbb{R}^n$ and $\overline{x} \in C$ are points that satisfy $|x-\overline{x}|=d(x,C)$, proves that $\langle x-\overline{x},y-\overline{x} \rangle \leq 0$ for all $y \in C$.

Homework Equations

By definition we have $d(x,C) = inf \{ |x-z| ; z \in C \}$.
And if $C$ is a convex set, then for any two points $v,w \in C$ we have $(1-t)v+tw \in C$.

The Attempt at a Solution

Due the fact that $|x-\overline{x}|=d(x,C)$ , I know that $|x-\overline{x}|^2 \leq |x-z|^2$ for all $z \in C$.
I tried to sketch a prove by contraction. So I supposed that there is an $y \in C$ that $\langle x-\overline{x},y-\overline{x} \rangle > 0$.
Then I used the fact that $C$ is a convex set, so $z=(1-t)y+t\overline{x} \in C$, where $t \in [0,1]$.
Then I intend to conclude something like $|x-z|^2 < |x-\overline{x}|^2$, what would be an absurd:

But the fact is I came nowhere with my attempt... =(

 

[fresh_42]

If I draw a picture, then the law of cosine and the triangle inequality will come to mind, especially as such an inner product is part of the law of cosine.

 

[tnich]

Homework Statement

Let $C \subset \mathbb{R}^n$ a convex set. If $x \in \mathbb{R}^n$ and $\overline{x} \in C$ are points that satisfy $|x-\overline{x}|=d(x,C)$, proves that $\langle x-\overline{x},y-\overline{x} \rangle \leq 0$ for all $y \in C$.

Homework Equations

By definition we have $d(x,C) = inf \{ |x-z| ; z \in C \}$.
And if $C$ is a convex set, then for any two points $v,w \in C$ we have $(1-t)v+tw \in C$.

The Attempt at a Solution

Due the fact that $|x-\overline{x}|=d(x,C)$ , I know that $|x-\overline{x}|^2 \leq |x-z|^2$ for all $z \in C$.
I tried to sketch a prove by contraction. So I supposed that there is an $y \in C$ that $\langle x-\overline{x},y-\overline{x} \rangle > 0$.
Then I used the fact that $C$ is a convex set, so $z=(1-t)y+t\overline{x} \in C$, where $t \in [0,1]$.
Then I intend to conclude something like $|x-z|^2 < |x-\overline{x}|^2$, what would be an absurd:
View attachment 230255

But the fact is I came nowhere with my attempt... =(

I think you are close here. I suggest writing the convexity condition as $z=(1-t)\overline{x}+ty \in C$, that is, swapping $t$ and $(1-t)$. This gives you points close to $\overline{x}$ when $t$ is small, and that is where you are likely to find a point $z$ that violates the condition $|x-\overline{x}|=inf \{ |x-z| ; z \in C \}$. Then, if you work through your derivation with this change, you get $\langle x-\overline{x},y-\overline{x} \rangle$ showing up explicitly and you can find a condition on $t$ that gives you what you need.

 

[Ray Vickson]

Homework Statement

Let $C \subset \mathbb{R}^n$ a convex set. If $x \in \mathbb{R}^n$ and $\overline{x} \in C$ are points that satisfy $|x-\overline{x}|=d(x,C)$, proves that $\langle x-\overline{x},y-\overline{x} \rangle \leq 0$ for all $y \in C$.

Homework Equations

By definition we have $d(x,C) = inf \{ |x-z| ; z \in C \}$.
And if $C$ is a convex set, then for any two points $v,w \in C$ we have $(1-t)v+tw \in C$.

The Attempt at a Solution

Due the fact that $|x-\overline{x}|=d(x,C)$ , I know that $|x-\overline{x}|^2 \leq |x-z|^2$ for all $z \in C$.
I tried to sketch a prove by contraction. So I supposed that there is an $y \in C$ that $\langle x-\overline{x},y-\overline{x} \rangle > 0$.
Then I used the fact that $C$ is a convex set, so $z=(1-t)y+t\overline{x} \in C$, where $t \in [0,1]$.
Then I intend to conclude something like $|x-z|^2 < |x-\overline{x}|^2$, what would be an absurd:
View attachment 230255

But the fact is I came nowhere with my attempt... =(

What theorems about convex sets have already been proved in your course and so are available for you to use? For example, have you seen the "separating hyperplane" theorem?

 

[Onezimo Cardoso]

Finally I done this question!
Thanks for all the help. I could do it following the tip tnich gave to me.
Soon I'll organize the solution and I'll post here as well.

 

[Onezimo Cardoso]

As I promisse, follow a solution for this question.

Let us suppose, by contraction, there is $y \in C$ such that $\langle x-\overline{x}, y - \overline{x} \rangle > 0$.

Let $z=ty+(1-t)\overline{x}$, where $t \in (0,1)$.

By hypothesis, $C$ is convex. Therefore, $z \in C$.

We have:

Then,

By the other hand, we have:

By hypothesis, $\langle x-\overline{x} , y - \overline{x} \rangle > 0$. Therefore, $| \langle x-\overline{x} , y - \overline{x} \rangle | = \langle x-\overline{x} , y - \overline{x} \rangle$. Then we have:

Thus,

Taking $t \in I^{*}$, we have:

Applying $(II)$ in $(I)$, we have:

Absurd!

Therefore,

 

[Onezimo Cardoso]

Another pretty good exercise it to use the exercise above to solve the following one:


Exercise: Let $C \subset \mathbb{R}^n$ be a convex and closed set. Let $f : \mathbb{R}^n \to C$ be a function defined as $f(x)=\overline{x}$, where $\overline{x}$ is the unique point of $C$ such that $|x-\overline{x}|=d(x,C)$. Prove that $|f(x)-f(y)|\leq |x-y|$ for any $x,y \in \mathbb{R}^n$, i.e., $f$ is uniformly continuous.

I could solve this one as well. I'll let it as an exercise, but if someone tries to solve it and cannot find a way I can post my solution here. Thanks guys!

 

[tnich]

Another pretty good exercise it to use the exercise above to solve the following one:


Exercise: Let $C \subset \mathbb{R}^n$ be a convex and closed set. Let $f : \mathbb{R}^n \to C$ be a function defined as $f(x)=\overline{x}$, where $\overline{x}$ is the unique point of $C$ such that $|x-\overline{x}|=d(x,C)$. Prove that $|f(x)-f(y)|\leq |x-y|$ for any $x,y \in \mathbb{R}^n$, i.e., $f$ is uniformly continuous.

I could solve this one as well. I'll let it as an exercise, but if someone tries to solve it and cannot find a way I can post my solution here. Thanks guys!

Looks good except for some minor errors (Cauchy - Schwartz inequality would require a $\leq$ sign, $|y-\overline x|^2$ in the denominator). But there are a couple of major ones as well.
1) you have only completed the proof for $x \notin C$. When $x\in C$, $\langle x-\overline{x},y-\overline{x} \rangle = 0$ and the chain of inequalities fails. Fortunately, it is easy to prove this special case.
2) Since $\overline x \in C$, $y$ can be as close to $\overline x$ as you like, meaning that $|x- \overline x| \leq |y - \overline x|$ is not true in general. The proof is still salvageable, though.

 

[tnich]

Looks good except for some minor errors (Cauchy - Schwartz inequality would require a $\leq$ sign, $|y-\overline x|^2$ in the denominator). But there are a couple of major ones as well.
1) you have only completed the proof for $x \notin C$. When $x\in C$, $\langle x-\overline{x},y-\overline{x} \rangle = 0$ and the chain of inequalities fails. Fortunately, it is easy to prove this special case.
2) Since $\overline x \in C$, $y$ can be as close to $\overline x$ as you like, meaning that $|x- \overline x| \leq |y - \overline x|$ is not true in general. The proof is still salvageable, though.

Try solving this inequality directly for $t$:
$-2t\langle x-\overline{x} , y - \overline{x} \rangle + t^2|y-\overline x|^2 <0$