Integral optimal flat-norm decompositions in codimension one
Let $d \ge 1$ and let $T$ be a compactly supported integral $d$-current in $\mathbb{R}^{d+1}$, so that $T$ and $\partial T$ are integer-multiplicity rectifiable currents of finite mass. With $\mathbf{M}$ the mass, the real flat norm of $T$ is $$\mathbb{F}(T) = \inf\{\mathbf{M}(X) + \mathbf{M}(S) : T = X + \partial S,\ X \in \mathcal{N}_d^c,\ S \in \mathcal{N}_{d+1}^c\},$$ where $\mathcal{N}_k^c$ is the space of compactly supported real normal $k$-currents in $\mathbb{R}^{d+1}$. Ibrahim, Krishnamoorthy and Vixie proved that every integral $1$-current in the plane has an integral optimal decomposition, and that every integral $d$-current in $\mathbb{R}^{d+1}$ has one provided simplicial complexes admit subdivisions of the kind in their Conjecture 3.4; they note that the case of $d$-currents that are boundaries of $(d+1)$-currents follows from results on the $L^1$ total variation problem, and that in codimension three or higher an optimal decomposition cannot in general be taken integral.
Problem. For every $d \ge 1$ and every such $T$, must there exist compactly supported integral currents $X_I$ and $S_I$ such that $$T = X_I + \partial S_I, \qquad \mathbb{F}(T) = \mathbf{M}(X_I) + \mathbf{M}(S_I)?$$ The input is not required to satisfy $\partial T = 0$. At least one integral optimum of the real problem is asked for; integrality of every optimum, and optimality only within the integral class, are different questions.
Source: Ibrahim, Krishnamoorthy and Vixie (2016), §1 and Theorem 3.7, where it is proved under their subdivision Conjecture 3.4. Stated by AIM — Open Applied Problems for compactly supported currents, without that hypothesis.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- MG
- Keywords
- AIM Open Applied Problems, flat norm, integral currents
References
@article{ibrahim2016,
author = {Ibrahim, Sharif and Krishnamoorthy, Bala and Vixie, Kevin R.},
title = {Flat norm decomposition of integral currents},
journal = {J. Comput. Geom.},
volume = {7},
number = {1},
year = {2016},
pages = {285--307},
doi = {10.20382/jocg.v7i1a14},
eprint = {1411.0882},
archivePrefix = {arXiv},
}
@misc{aim-integral-flat-norm-codimension-one,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Integral optimal flat-norm decompositions in codimension one}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/501-integral-flat-norm-codimension-one.md},
note = {Entry 501 at commit 24edaad, 2026-10-15},
}
Resolution claims
The real flat norm decomposes a current into a residual and the boundary of a filling, minimizing their total mass. We prove that every compactly supported integral $d$-current in $\R^{d+1}$ has an integral optimal decomposition, including when its boundary is nonzero. The filling is a scalar function of bounded variation. Rounding its values with one common shift preserves the mean …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/integral_flat_norm.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00801, "Integral optimal flat-norm decompositions in codimension one", clopen.solutions.