Structure of Minimal Locally Concave Functions

@misc{dobronravov2026structure,
  title = {{Structure of Minimal Locally Concave Functions}},
  author = {Egor Dobronravov},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.08783},
  eprint = {2608.08783},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.08783},
}

TY  - UNPB
TI  - Structure of Minimal Locally Concave Functions
AU  - Egor Dobronravov
PY  - 2026
DA  - 2026/08/09
PB  - arXiv
DO  - 10.48550/arxiv.2608.08783
UR  - https://arxiv.org/abs/2608.08783
AB  - We propose a theory describing the structure of minimal locally concave functions on an arbitrary subdomain of ℝ^d. We introduce the notion of an extremal set, that is, a set on which the function is affine, and prove that extremal sets foliate the domain. We show that extremal sets are covered by simplices with vertices on the boundary of the domain, and that these simplices themselves lie in the extremal set. We also prove that one-dimensional extremal sets can approach the free boundary only tangentially, not transversally.
ER  -