**Monday, December 18, 2017 **

Technische Universität Berlin

Institut für Mathematik

Straße des 17. Juni 136

10623 Berlin

room MA 041

** Lecture - 14:15**

*Abstract:*

It is often assumed that the only way to represent a (compact) non-convex
polyhedron $X\subset \RR^d$ is by partitioning into convex polytopal pieces.
Such partitions are very far from being unique or succinct; in particular in
dimensions $d>2$ one might need to introduce {\sl Steiner points}---points in
the inerior of $X$ which must arise as vertices of polytopal pieces.

We propose a description that is essentially unique: a rational function
associated with the uniform measure $\mu(X)$ supported on $X$. It is given
by an integral transform of $\mu(X)$ known as {\sl Fantappi\`e transformation}
$${\cal F}(\mu(X))(u):=
d!\int_{\RR^{d}} {d\mu(X)(x)\over (1-\la u,x\ra)^{d+1}}={P(u)\over
\prod_{v\in V}(1-\la u,v\ra)},$$
where $\la x,y\ra$ denotes the standard scalar product $\sum_i x_i y_i$.
Expanding ${\cal F}$ into Taylor series gives a scaled
moment generating function (m.g.f.) for $\mu(X)$, known in the univariate case as factorial m.g.f.,
and this allows for efficient computation of integrals over $X$, etc.

${\cal F}$ admits decompositions into ``elementary''
summands with real weights, corresponding to certain simplices with vertices
from $V$, which are analogous to triangulations of convex polytopes. In
particular they allow an efficient way to check containment of a point in $X$.
It is an interesting open question whether these weights can be chosen to be
$\pm 1$.

It is natural to homogenise ${\cal F}$, an operation that corresponds to
switching to the Laplace-Fourier transform of a measure supported on the conic
closure of $X$ embedded into the hyperplane $\{x_0=1\}\subset\RR^{d+1}$. It
allows to apply powerful machinery of Jeffrey-Kirwan residues by Brion and
Vergne, used in the theory of hyperplane arrangements.

joint work with N. Gravin, B. Shapiro and M. Shapiro

* * *

For a PDF version please click
** here **

**Colloquium - 16:00**

*Abstract:*

In this note we study the Banach-Mazur distance between the n-dimensional cube and the crosspolytope. Previous work shows that the distance has order $\sqrt{n}$ , and here we will prove some explicit bounds improving on former results. Even in dimension 3 the exact distance is not known, and based on computational results it is conjectured to be $\frac{9}{5}$. Here we will also present computerbased potential optimal results in dimension 4 to 8.

Letzte Aktualisierung:
15.12.2017