Can I talk about a non-linear shape functional. I understand a shape functional $J$ as some mapping that takes a shape and returns a real (or complex) value.
I would like to talk about a non-linear shape functional as one that does not fulfill:
$$ J\left(A\cup B\right)=J\left(A\right)+J\left(B\right) $$ where $A$ and $B$ are two subsets of the set $\Omega$.
The problem here is that, denoting $\mathcal{P}\left(\Omega\right)$ as the power set of $\Omega$ (i.e. the collection of its subsets), $\left(\mathcal{P}\left(\Omega\right),\cup\right)$ only has monoid structure, but not a vector space structure.
So really what i'm saying is that $J:\mathcal{P}\left(\Omega\right)\to\mathbb{R}$ does not conserve the monoid structure. However I always hear people saying "it's not linear" so I would like to know it it makes any sense to refer it like that, of if I'm missing something...
If someone can point me to some useful reference that would be also fine.