- #1

Calabi

- 140

- 2

## Homework Statement

**Hello. Let be n integer and let's considere $$\mathbb{R}^{n} = E$$.**

Let be the distance $$d : (x; y) \in \mathbb{R}^{2} \rightarrow ||x - y||_{2}$$.

We wright $$\forall p \in [1; \infty], B_{p}$$ the unit closed ball for $$||.||_{p}$$.

Forall compacts A, B of $$E$$, we define $$d'(A; B) = sup_{x \in \mathbb{R}^{n}} |inf_{y \in A}d(x; y) - inf_{y \in B}d(x; y)|$$.

My goal is to show tha forall $$p_{0} \in [1; +\infty]$$, $$d'(B_{p_{0}}, B_{p}) \rightarrow_{p \rightarrow p_{0}} 0$$.

Let be the distance $$d : (x; y) \in \mathbb{R}^{2} \rightarrow ||x - y||_{2}$$.

We wright $$\forall p \in [1; \infty], B_{p}$$ the unit closed ball for $$||.||_{p}$$.

Forall compacts A, B of $$E$$, we define $$d'(A; B) = sup_{x \in \mathbb{R}^{n}} |inf_{y \in A}d(x; y) - inf_{y \in B}d(x; y)|$$.

My goal is to show tha forall $$p_{0} \in [1; +\infty]$$, $$d'(B_{p_{0}}, B_{p}) \rightarrow_{p \rightarrow p_{0}} 0$$.

## Homework Equations

$$d'(A; B) = sup_{x \in \mathbb{R}^{n}} |inf_{y \in A}d(x; y) - inf_{y \in B}d(x; y)|$$[/B]

**I also simplify this distance by showing $$d'(A; B) = max (sup_{x \in A} d(x; B), sup_{x \in B}d(x; A))$$.**

## The Attempt at a Solution

**At the moment I just think about the case where n = 2 : the sup is reach I think**

is reach on the $$y = x$$ space.

I also ask my self about the inequality between norm.

is reach on the $$y = x$$ space.

I also ask my self about the inequality between norm.

**What do you think please?**

**Thank you in advance and have a nice afternoon.**