**1. Projection and slicing in the Heisenberg group **

with K. Fässler, P. Mattila, J. Tyson.

We aim at studying complicated objects in metric spaces such as Euclidean space or Heisenberg group. Tomography is a possible approach : cut 2D slices in a 3D object.

** 1.1. Euclidean projection and slicing **

In 1954, Marstrand observed that given a planar set of Hausdorff dimension , its orthogonal projection to the line of polar angle , cannot be small for all .

**Theorem 1 (Marstrand 1954)** * For almost every ,
*

*
** Furthermore, in the second case, . *

de Saxcé: there are examples of sets with for which there is a dense of ‘s such that .

**Theorem 2 (Kaufman-Mattila 1975)** * Let be a subset of . Let be an integer between 1 and . For almost every orthogonal projection of rank ,
*

*
** Furthermore, in the second case, . *

de Saxcé: This holds for a rather large class of measures on the Grassmannian, satisfying a Hölder type assumption.

A bit more is true: not only does intersect for a set of ‘s of positive measure in .

**Theorem 3 (Mattila’s Slicing Theorem)** * Let . For a.e. -dimensional subspace , for a set of ‘s of positive measure in , the intersection of with has dimension . *

** 1.2. Projection and slicing in the Heisenberg group **

The role of will be played by horizontal isotropic subspaces, contains the center. These are subgroups, with a normal subgroup. is a semi-direct product. The projection is a group homomorphism, it is 1-Lipschitz. On the other hand, is merely -Hölder continuous. The horizontal isotropic Grassmannian is a Riemannian homogeneous space.

**Theorem 4** * let A be a Borel subset of Heisenberg group. For a.e. -dimensional horizontal isotropic , (easy) but *

*
*
- .
- Furthermore, if , .
- Let . Then, for a set of ‘s of positive measure in , .

* These bounds are sharp: every pair allowed by these inequalities is achieved. *

**Remark 1** * In particular, in , we have the projection property for a sub-Grassmannian, the isotropic Grassmannian, which is a rather small subspace of the full Grassmannian of -planes in . *

de Saxcé: doesn’t it follow from Kaufmann’s estimate on the dimension of the set of bad projections ? Answer: no, Kaufmann’s upper bound is larger than the dimension of the isotropic Grassmannian

**Theorem 5** * (easy) but*

*
*
For a.e. -dimensional horizontal isotropic ,

* *

We do not know what happens when , except for an upper bound in the first Heisenberg group.

Proof relies on the energy characterization of dimension (equality of Hausdorff and capacity dimensions).

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## About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See
http://www.math.ens.fr/metric2011/