# 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

#### 5 Citations

Precisely monotone sets in step-2 rank-3 Carnot algebras

- Mathematics
- 2021

A subset of a Carnot group is said to be precisely monotone if the restriction of its characteristic function to each integral curve of every left-invariant horizontal vector field is monotone.… Expand

Horizontally affine maps on step-two Carnot groups

- Mathematics
- 2020

In this paper we introduce the notion of horizontally affine, $h$-affine in short, maps on step-two Carnot groups. When the group is a free step-two Carnot group, we show that such class of maps has… Expand

Polynomial and horizontally polynomial functions on Lie groups

- Mathematics
- 2020

We generalize both the notion of polynomial functions on Lie groups and the notion of horizontally affine maps on Carnot groups. We fix a subset $S$ of the algebra $\mathfrak g$ of left-invariant… Expand

Multiexponential maps in Carnot groups with applications to convexity and differentiability

- Mathematics
- 2019

We analyze some properties of a class of multiexponential maps appearing naturally in the geometric analysis of Carnot groups. We will see that such maps can be useful in at least two interesting… Expand

Horizontally affine functions on step-2 Carnot algebras

- Mathematics
- 2020

In this paper we introduce the notion of horizontally affine, h-affine in short, function and give a complete description of such functions on step-2 Carnot algebras. We show that the vector space of… Expand

#### References

SHOWING 1-8 OF 8 REFERENCES

Regularity of sets with constant horizontal normal in the Engel group

- Mathematics
- 2012

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

- Mathematics
- 2005

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

- Mathematics
- 2015

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 minimizing… Expand

Twisted convex hulls in the Heisenberg group

- Mathematics
- 2007

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 twisted… Expand

Metric differentiation, monotonicity and maps to L1

- Mathematics
- 2009

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

- Mathematics
- 2012

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 indeed… Expand

Rigot, Semmes surfaces and intrinsic Lipschitz graphs in the Heisenberg group, ArXiv e-prints (2018)

- 2018