show/hide this revision's text 3 Responded to Pietro Majer's comments

This may be an ill-posed question, but suppose I have a collection of continuous, bounded, scalar-valued nonnegative functions $f_1(x,y),\dots,f_n(x.y)$ defined on the closed unit disk. Given a collection of scalars $a_1,\dots,a_n$, define

$R_i = \{(x,y) : f_i(x,y) + a_i \leq f_j(x,y) + a_j \ \forall j \ \} $

for each index $i$. Are there any sufficient conditions I can impose on these functions that will guarantee that the $R_i$ are connected for all $a_i$? It works if the functions are linear, for example (since the $R_i$ end up being convex), but I'd like something as general as possible.
show/hide this revision's text 2 TeX

This may be an ill-posed question, but suppose I have a collection of scalar-valued nonnegative functions $f_1(x,y),\dots,f_n(x.y)$ defined on the unit disk. Given a collection of scalars $a_1,\dots,a_n$, define

$R_i = {(x,y) \{(x,y) : f_i(x,y) + a_i \leq f_j(x,y) + a_j \ \forall j \ \} $

for each index $i$. Are there any sufficient conditions I can impose on these functions that will guarantee that the $R_i$ are connected for all $a_i$? It works if the functions are linear, for example (since the $R_i$ end up being convex), but I'd like something as general as possible.
show/hide this revision's text 1

Connected level sets

This may be an ill-posed question, but suppose I have a collection of scalar-valued nonnegative functions $f_1(x,y),\dots,f_n(x.y)$ defined on the unit disk. Given a collection of scalars $a_1,\dots,a_n$, define

$R_i = {(x,y) : f_i(x,y) + a_i \leq f_j(x,y) + a_j \forall j } $

for each index $i$. Are there any sufficient conditions I can impose on these functions that will guarantee that the $R_i$ are connected for all $a_i$? It works if the functions are linear, for example (since the $R_i$ end up being convex), but I'd like something as general as possible.