What does the notation $\sup_{x\in X}\inf_{y\in Y}d(x,y)$ mean?

Question

Say you have two disjoint closed sets $X$ and $Y$ in a metric space. I'm trying to interpret what $$\sup_{x\in X}\inf_{y\in Y}d(x,y)$$ means.

Does it mean that you pick a fixed $x\in X$, and then compute $\inf_{y\in Y}d(x,y)$ for this fixed $x$ as $y$ ranges over $Y$, to get a set of infimums, one for each $x\in X$, and then take the supremum of that resulting set?

score 2 · Accepted Answer · answered Aug 17 '13 at 01:22

2

You can unravel this in the following way:

For every $x\in X$ let $y_x = \inf\{d(x,y)\mid y\in Y\}$.
Now we take $\sup\{y_x\mid x\in X\}$.

answered Aug 17 '13 at 01:22

Asaf Karagila

393,674

score 0 · Answer 2 · answered Aug 17 '13 at 01:20

0

Exactly as you said.${}{}{}{}{}$

answered Aug 17 '13 at 01:20

Lord Soth

7,750
20
37

What does the notation $\sup_{x\in X}\inf_{y\in Y}d(x,y)$ mean?

2 Answers2