# Workshop "Busemann-Petty problems"

## Program

The goal of this workshop is to revive our interest in the research program set up by Busemann and Petty in their 1956 paper “Problems on convex bodies”. Having its roots in the theory of Minkowski spaces, it led to various fundamental questions on the geometry of convex bodies, most of which remain unsolved.

Mini-course by Juan-Carlos Alvarez Paiva (Lille) “The Busemann-Petty Problems: one down, nine to go” / “Les problèmes de Busemann-Petty: un de fait, neuf à faire”.

Abstract: In their 1956 paper Busemann and Petty formulated ten interesting and difficult problems on the theory of convex bodies. These problems originated from their investigations on the notions of volume and area on normed spaces and their natural generalization, Finsler spaces. Of the ten problems only the first has been completely solved and its solution shed much light on the role of the Fourier transform and the theory of distributions on convex geometry. These talks will be an introduction to the theory of volumes and areas on normed and Finsler spaces with the primary purpose of placing the Busemann-Petty problems (and their duals) in their natural context. A secondary aim of the talks is to exhibit the interplay between integral geometry, symplectic geometry, variational calculus, and the geometr! y of numbers of which the Busemann-Petty problems are an example.

Lecture by M. Fradelizi: “Solution of the first Busemann-Petty problem and Busemann's inequality”

Abstract: The first of the ten Busemann-Petty problems asks the following: if two symmetric convex bodies $K$ and $L$ in $\mathbb{R}^n$ satisfy $|K\cap u^\bot|_{n-1}\le |L\cap u^\bot|_{n-1}$ can one deduce that $|K|_n\le|L|_n$? We shall present the works of Lutwak, Koldobsky, Zhang and Gardner which gave the solution of the problem (yes for $n\le4$ and no for $n\ge5$), then the extension of these results to other measures by Zvavitch. We shall also show how the generalization of Brunn's principle to convex measures by Borell enables to prove very easily (an extension of) Busemann's inequality on the concavity of the volume of sections of convex bodies by hyperplanes through the origin.

Lecture by D. Cordero-Erausquin: “Minimality of planes in normed spaces, after Burago and Ivanov”

Abstract: Busemann proved that the measure of hyperplane sections of a convex body verify some form of convexity with respect to the normal direction of the hyperplanes. He asked whether the same property (which needs first to be properly stated) is also true for lower dimensional sections. This is still open, except for the case of 2-dimensional sections which was recently positively solved by Burago and Ivanov. I will try to present their proof, which relies on the notion of calibration.

Lecture by G. Berck: “Convexity of Intersection Bodies”

Abstract: The tenth Busemann-Petty problem asks wether flat subspaces of a norm space are area minimizers for the Haussdorff measure. For a long time, only the hyperplanes were known to have minimal area and this is a consequence of Busemann's theorem on the convexity of intersection bodies. In this talk, we wil give two proofs of this theorem, one geometric with the help of floating bodies and one analytical which directly extends to a proof of the convexity of L_p-intersection bodies.
Slides of the lecture

Lecture by A. Bernig: “Minimality of $k$-planes in normed spaces”

Abstract: We introduce a new volume definition on normed vector spaces. We show that the induced $k$-area functionals are convex for all $k$. In the particular case $k=2$, our theorem implies that Busemann's 2-volume density is convex, which was recently shown by Burago-Ivanov. We also show how the new volume definition is related to the centroid body and prove some affine isoperimetric inequalities.

Lecture by C. Vernicos “Projective invariants of convex sets and projective inequalities”

Abstract: We will present different invariants arising from the study of the Hilbert and Funk structures of convex sets, and the state of the art on the various conjectures associated with them.

Time for informal discussions

## Participants

Participants Affiliation
Barthe, Franck Université de Toulouse
Alvarez Paiva, Juan-Carlos Université de Lille 1
Cordero-Erausquin, Dario Université Pierre et Marie Curie (P6)
Huou, Benoit Université Paul Sabatier, Toulouse
Bouyrie, Raphaël Université Paul Sabatier, Toulouse
Thomas, Pascal IMT
Bernig, Andreas Goethe University Frankfurt
Lehec, Joseph Université Paris-Dauphine
Yassine, Zeina Université Paris Est Créteil (UPEC)
Abardia, Judit Goethe University Frankfurt
Meckes, Mark Case Western Reserve University
Berck, Gautier P&V
Bertrand, Jérôme IMT
Vritsiou, Beatrice-Helen Université Pierre et Marie Curie
Vernicos, Constantin Université Montpellier 2
Delplancke, Claire Université Paul Sabatier, Toulouse
Meckes, Elizabeth Case Western Reserve University
Guédon, Olivier Université Paris-Est Marne La Vallée
Meyer, Mathieu Université Paris-Est Marne-la-Vallée

## Schedule

Slot\Day Mon 13 Tue 14 Wed 15
9:30-10:00 J.C. Alvarez Paiva (3/3)
10:00-10:30J.C. Alvarez Paiva (2/3)
10:30-11:00
11:00-11:30 A. Bernig
12:00-13:30 Lunch
13:30-14:00 Welcome G. Berck
14:00-14:30J.C. Alvarez Paiva (1/3)
14:30-15:00 Discussion
15:00-15:30 Work in groups
15:30-16:00 Coffee
16:30-17:00 Coffee
17:00-17:30 C. Vernicos D. Cordero
17:30-18:00

## Practical information

The workshop will take place at the Institute of Mathematics of Toulouse, building 1R1, room 106.

How to get there: take Metro line B towards Ramonville, exit at “Université Paul Sabatier” and walk 150m. More details.