2
$\begingroup$

Let $$\chi_S(x,y)=\begin{cases}1&\text{ if }0< x<y< 1\\0&\text{ else }\end{cases}$$

be the indicator function of the simplex $S=\{(x,y)\in (0,1)^2:x<y\}$. I am interested in an explicit partition of unity. That is, I am looking for continuous positive functions $f_n:\mathbb R^2\to \mathbb R$ so that $\sum_{n=1}^\infty f_n(x,y)=\chi_S(x,y).$

Does anyone have a reference to such an example? I suspect this should be relatively standard but I have been unable to find anything.

Improve this question
$\endgroup$
5
  • $\begingroup$ Such a partition does not exist, because $\sum_{n=1}^\infty f_n$ must be $0$ on the boundary of $S$. $\endgroup$
    – Iosif Pinelis
    Commented Jul 11, 2023 at 18:44
  • $\begingroup$ Look at Theorem 3.11 in M. Spivac, Calculus on Manyfolds. $\endgroup$
    – user64494
    Commented Jul 11, 2023 at 18:44
  • $\begingroup$ @IosifPinelis Thank you. I will edit, I mean the indicator of the open simplex. $\endgroup$
    – user479223
    Commented Jul 11, 2023 at 18:45
  • $\begingroup$ @user64494 This gives existence of a partition of unity - I would like an explicit example if possible $\endgroup$
    – user479223
    Commented Jul 11, 2023 at 18:51
  • $\begingroup$ @user479223: It is easy to derive an explicit partition from that theorem. Good luck! $\endgroup$
    – user64494
    Commented Jul 11, 2023 at 19:10

1 Answer 1

Reset to default
3
$\begingroup$

It is easy to construct an explicit continuous (or even smooth) partition of unity over $\mathbb R^2$. Map this partition of unity to the continuous (or smooth) partition of the indicator $\chi_Q$ of the open square $Q:=(-1,1)^2$ via the bi-smooth map $\mathbb R^2\ni(x,y)\mapsto\frac2\pi\,(\arctan x,\arctan y)\in Q$. Finally, use an explicit homeomorphism from (open convex set) $Q$ onto (open convex set) $S$ to get an explicit continuous partition of $\chi_S$.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .