I have a formula containing two squared root components as follows:
$1 - \frac{\sqrt{\sum_{\forall i \in \mathcal{I}}\sum_{\forall j \in \mathcal{J}}(a_{ij} - b_{ij})^2}}{\sqrt{\sum_{\forall i \in \mathcal{I}}\sum_{\forall j \in \mathcal{J}}a_{ij}^2}}$.
Assume that $a_{ij}, \forall i \in \mathcal{I}, \forall j \in \mathcal{J}$ are constants. How to know that the above equation is concave or convex?
We can view it as $$1-\frac1u \|A-B\|_F$$ where $u$ is a positive constant.
since we know that the frobenius norm is convex, multiplying by a negative number makes it concave. Shifting it by $1$ doesn't change the property.
Hence it is a concave function.