On the inner cone property for convex sets in two-step Carnot groups, with applications to monotone sets

@article{Morbidelli2018OnTI,
  title={On the inner cone property for convex sets in two-step Carnot groups, with applications to monotone sets},
  author={D. Morbidelli},
  journal={arXiv: Metric Geometry},
  year={2018}
}
In the setting of step two Carnot groups, we show a "cone property" for horizontally convex sets. Namely we prove that, given a horizontally convex set $C$, a pair of points $P\in \partial C$ and $Q\in $ int $C$, both belonging to a horizontal line $\ell$, then an open truncated subRiemannian cone around $\ell$ and with vertex at $P$ is contained in $C$. We apply our result to the problem of classification of horizontally monotone sets in Carnot groups. We are able to show that monotone sets in… Expand
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References

SHOWING 1-8 OF 8 REFERENCES
Regularity of sets with constant horizontal normal in the Engel group
In the Engel group with its Carnot group structure we study subsets of locally finite subRiemannian perimeter and possessing constant subRiemannian normal. We prove the rectifiability of such sets:Expand
Geodetically Convex Sets in the Heisenberg Group
Recently, several notions of convexity have been introduced and studied in Heisenberg groups and in more general Carnot groups. A weak and a strong definition of convex function are discussed in [4]:Expand
On the lack of semiconcavity of the subRiemannian distance in a class of Carnot groups
We show by explicit estimates that the SubRiemannian distance in a Carnot group of step two is locally semiconcave away from the diagonal if and only if the group does not contain abnormal minimizingExpand
Twisted convex hulls in the Heisenberg group
We define and investigate the notion of twisted convex hull for a subset of the Heisenberg group IH. We show that, while the twisted convex hull of two points is always a bounded set, the twistedExpand
Metric differentiation, monotonicity and maps to L1
This is one of a series of papers on Lipschitz maps from metric spaces to L1. Here we present the details of results which were announced in Cheeger and Kleiner (Ann. Math., 2006, to appear,Expand
Regularity Properties of H-Convex Sets
We study the first- and second-order regularity properties of the boundary of H-convex sets in the setting of a real vector space endowed with a suitable group structure: our starting point is indeedExpand
Rigot, Semmes surfaces and intrinsic Lipschitz graphs in the Heisenberg group, ArXiv e-prints (2018)
  • 2018