In this post, I give a possible definition for step functions in and a related problem and its possible solution.
Let be the arc length parametrization of a plane, smooth, closed, convex curve, of length . Let be a smooth and Bounded variation function, with . Let be the set of all interior points of the region bounded by the curve .
The problem is to determine , such that
1. , for all not in the inetrior of .
2. Given any , and let ,
3. , where is a set such that , meaning 1-Hausdorff measure of is 0.
4. The total variation is minimum possible, where
and is the weak/distributional gradient of .
Motivation : is my definition of a step function in
I roughly sketch a solution without a proof, and also a possible minimum possible value for total variation of .
Let assume values of on the boundary curve . We construct infinite number of scaled down versions of the curve with scale factor ranging from and converging to , and on each curve we let take values of corresponding scaled down version of . If curve is scaled by , then new function is taken as
where is defined on .
This is illustrated roughly in the picture below.
This way we construct on entire and it is smooth(as and are smooth) except possibly at , the centroid of region , where it is not continuous. But this discontinuty at has no effect on Variation of in .
Variation of for this case, is given as
I am yet to sketch a proof that this is the minimum possible variation for .
PS : is the total variation of in , and is the two dimensional Hausdorff measure of the set .