- #1

- 34

- 0

Please note that f(x)= inf_{y∈F} d(x,y) means that y∈F is written below the inf...it is hard to show it on one line, so I chose to show as a subscript...Can anyone tell me how I should begin this problem?

You should upgrade or use an alternative browser.

In summary, the conversation discusses the problem of showing that the function f(x) defined as the infimum of the distances between x and a non-empty subset F of a metric room (X,d) is continuously uniform in the entire X. The conversation also presents a general solution for this problem, which involves proving a stronger inequality. The key insight is to consider the special case where F consists of only two points, and then generalize the solution to the case of arbitrary F.

- #1

- 34

- 0

Please note that f(x)= inf_{y∈F} d(x,y) means that y∈F is written below the inf...it is hard to show it on one line, so I chose to show as a subscript...Can anyone tell me how I should begin this problem?

Physics news on Phys.org

- #2

Science Advisor

Homework Helper

Gold Member

- 4,809

- 31

- #3

Science Advisor

Homework Helper

Gold Member

- 4,809

- 31

The general solution I found is as follows. Tell me what you think.

First, note that we can assume without loss of generality that d(y,F)≤d(x,F) [if not, interchange the labels of x and y...]. In this case, |d(x,F)-d(y,F)|=d(x,F)-d(y,F) and so we must prove that d(x,F)≤d(x,y)+d(y,F) [i.e., we must prove a kind of triangle inequality in which sets are allowed to occupy one of the variable position].

Next, consider {e_n}, {f_n} two sequences of points in F such that d(x,F)=lim d(x,e_n) and d(y,F)=lim d(y,f_n) and further assume that d(x,e_n)≤d(x,f_n) for all n in

Then, we have d(x,y)+d(y,f_n) ≥ d(x,f_n) ≥ d(x,e_n) for all n in

Share:

- Replies
- 26

- Views
- 567

- Replies
- 9

- Views
- 515

- Replies
- 3

- Views
- 947

- Replies
- 8

- Views
- 608

- Replies
- 9

- Views
- 161

- Replies
- 3

- Views
- 393

- Replies
- 3

- Views
- 649

- Replies
- 3

- Views
- 214

- Replies
- 4

- Views
- 544

- Replies
- 1

- Views
- 507