The problem involves proving the existence of a minimum distance from a point ##p_0## to a nonempty closed subset ##S## of ##E^n##. The discussion centers around the properties of closed sets and the behavior of distance functions.
The discussion is ongoing, with participants raising questions about the definitions and properties of minimum distances in relation to closed sets. Some guidance has been offered regarding the nature of the distance set being non-empty and bounded below.
Participants note the distinction between closed and open sets, particularly in the context of the existence of minimum values, and question the implications of these properties on the problem at hand.
Lee33 said:Homework Statement
Prove that if ##S## is a nonempty closed subset of ##E^n## and ##p_0\in E^n## then ##\min\{d(p_0,p):p\in S\}## exists.
2. The attempt at a solution
If ##p_0## was in ##S## why would ##\min\{d(p_0,p):p\in S\} = 0?## Is it just because it is the minimum? How about if ##p_0 \in S## then what will ##\max\{d(p_0,p):p\in S\}## be?
Lee33 said:Homework Statement
Prove that if ##S## is a nonempty closed subset of ##E^n## and ##p_0\in E^n## then ##\min\{d(p_0,p):p\in S\}## exists.