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.



Partition of unity of simplex
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 $\chi\_S$.


