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\centerline{Analytic Aspects of Convex Geometry}

\centerline{Program}

\noindent 9:00-9:45 E.~Werner: {\em Affine surface areas and
approximation of convex bodies by polytopes}

\noindent 9:50-10:20 A.~Colesanti: {\em Convex bodies and partial
differential equations: problems in between}

\vspace{0.3cm}

\noindent
10:20-10:45 Coffee break

\vspace{0.3cm}

\noindent 10:50-11:05 P.~Goodey: {\em Mixed support functions}

\noindent 11:10-11:25 L.~Brandolini: {\em On the average decay of
the characteristic function of a convex body}

\noindent 11:30-11:45 E.~L.~Grinberg: {\em Inversion of the Radon
transform via G{\aa}rding-Gindikin fractional integrals}

\noindent 11:50-12:05 M.~Ferri: {\em Bounds on the size functions
of 3D convex objects from 2D projections}

\noindent 12:10-12:25 P.~M.~Gruber: {\em Approximating of convex
bodies in $\mathbb{E}^3$ by circumscribed polytopes}

\noindent 12:30-12:45 P.~Gronchi:{\em Revisited inequalities for
convex sets}

\vspace{0.3cm}

\noindent
13:00-13:45 Lunch

\vspace{0.3cm}

\noindent 15:00-15:45 G.~Bianchi: {\em Determining convex sets
from their covariogram and related problems}

\noindent 15:50-16:20 A.~Koldobsky: {\em The Busemann-Petty
problem via spherical harmonics}

\vspace{0.3cm}

\noindent
16:20-16:45 Coffee break

\vspace{0.3cm}

\noindent 16:50-17:05 R.~Howard: {\em Estimating the distance to
the convex hull}

\noindent 17:10-17:25 C.~Peri: {\em Point $X$-rays of convex
bodies in planes of constant curvature}

\noindent 17:30-17:45 D.~Klain: {\em The Minkowski problem for
polytopes}

\noindent 17:50-18:05 P.~Salani: {\em Minkowski sum of
quasi-convex functions and $p$-capacity of convex sets}

\noindent 18:10-18:25 R.~Schneider: {\em Stability results
involving surface area measures of convex bodies}

\noindent 18:30-18:45 C.~Zanco: {\em Tilings of normed spaces: The
state of the art}


\pagebreak


\centerline{\large \bf  Determining convex sets from
their covariogram and related problems }

%\centerline{\large \bf Title to be Continued Here if Too Long}

\bigskip

\centerline{Gabriele Bianchi}
\centerline{\it Universit\`a degli Studi di Firenze, Italy}
\centerline{e-mail {\tt gabriele.bianchi@unifi.it}}

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par
Let $K$ be a convex body in ${\mathbb R}^n$. The {\it covariogram} $g_K(x)$ of
$K$ is the function
$$
g_K(x)= V(K\cap(K+x)),
$$
where $x\in{\mathbb R}^n$ and $V$ denotes volume in ${\mathbb R}^n$. The
covariogram is clearly
unchanged by a translation or a reflection.
The following problem was posed by Matheron:
{\em Does the covariogram determine a convex body, among all convex bodies,
up to translation and reflection?}
\medskip

This problem has been posed independently also in other contexts, like in
probability and in stochastic geometry. The covariogram of $K$ is, up to a
multiplicative factor, the probability density of the difference of two
independent random variables
$X-Y$ which are uniformly distributed over $K$.
The knowledge of $g_K$ is equivalent to the knowledge of the length
distribution of the chords of $K$ parallel to $u$, for each direction
$u$.

The covariogram problem is also equivalent to the
phase retrieval problem in Fourier analysis, restricted to the
class of characteristic functions of convex bodies.
\medskip

We will present recent results obtained on this
problem. The answer is known to be positive for planar convex
bodies with piecewise $C^2$ boundary (plus an extra minor condition),
while it has been proved that in ${\mathbb R}^n$, $n\geq4$, there are
convex bodies which are not determined by their covariogram. There are
currently attempts in solving the problem both for polyhedrons and for
bodies with $C^2_+$ boundary in ${\mathbb R}^3$. \bigskip


\pagebreak


\centerline{\large \bf On the average decay of the characteristic}

\centerline{\large \bf function of a convex body}

\bigskip

\centerline{Luca Brandolini} \centerline{\it Universit\`a di
Bergamo ,Italy} \centerline{e-mail: {\tt brandolini@unibg.it}}

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par

An important question in the study of the Fourier transform of the
Lebesgue measure on a convex bounded set in ${\Bbb R}^d$ is
whether the $L^2$-average decay rate is optimal in the sense that
$$ {\left(\int_{S^{d-1}} {|\widehat{\chi}_B(R\omega)|}^2 d\omega
\right)}^{\frac{1}{2}} \leq C R^{-\frac{d+1}{2}}, $$ which is
exactly the point-wise decay rate in the case when $B$ is the
ball. This estimate has extensive applications in numerous
problems in harmonic analysis and convexity including the
distribution of lattice points in convex domains, irregularity of
distribution, generalized Radon transform and others. We present
some results that hold under minimal regularity assumption.


\pagebreak


\centerline{\large \bf Convex bodies and partial differential
equations:}

\centerline{\large \bf problems in between}

\bigskip

\centerline{Andrea Colesanti}
\centerline{Universit\`a degli Studi di Firenze -- Italy}
\centerline{e--mail: {\tt andrea.colesanti@math.unifi.it}}

%%% in case of several authors:
%\smallskip
%\centerline{Author }2
%\centerline{\it Institution Name,
%Country}
%\centerline{e-mail {\tt Author2}}
%%%

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
\par
%
I would like to present some problems, and related results, which
are connected with both the classic theory of convex bodies and
elliptic partial differential equations (PDE's) of second order.

The first problem is to prove (or disprove) that a smooth convex
function $u$ of $n$ variables, defined in $\mathbf{R}^n$, such
that one of the elementary symmetric functions $S_i(D^2u(x))$,
$i=1,2,\dots n$, of the Hessian matrix $D^2u (x)$ is equal to
some positive constant for every $x$, is a quadratic polynomial.
This problem may be regarded as a counterpart, in the field of
PDE's, of the following: prove that a convex body in
$\mathbf{R}^n$ such that one of its area measures is proportional
to the $(n-1)$--dimensional Hausdorff measure on $S^{n-1}$, is a
ball. The latter problem is completely settled, while the
corresponding one in PDE's is solved for $i=1$ and $i=n$. For
arbitrary $i$, I proved that the claim is true if a condition on
the growth of $u$ at infinity is added.

The second problem regards capacity of $n$--dimensional convex
bodies; this was studied by myself and P. Salani. C. Borell
proved that the classical electrostatic capacity, or 2--capacity,
of convex bodies satisfies a Brunn--Minkowski type inequality. We
generalized this result to the $p$--capacity, for $1<p\le n$. The
notion of $p$--capacity arises in the theory of Sobolev spaces,
and one of the equivalent definition involves the solution of an
elliptic PDE.

Finally, together with D. Hug, we proved the following result:
let $u$ be a convex function defined in $\mathbf{R}^n$, and
assume that the Hessian measure of order one of $u$ has a
$p$--integrable density with respect to the $n$--dimensional
Lebesgue measure, for some $p>1$. Then one can deduce absolute
continuity results on Hessian measures of higher order (depending
on $p$). This follows from classical results for functions whose
Laplacian is $p$--integrable. As a consequence we obtain a new
proof of a result by W. Weil, regarding absolute continuity of
area measures of convex bodies.

\bigskip

%%% in case of several authors:
%{\bf  Presented by ...}
%%%


\pagebreak


\centerline{\large \bf Bounds on the size functions of 3D convex
objects}

\centerline{\large \bf from 2D projections}

\bigskip

\centerline{Massimo Ferri}
\centerline{\it ARCES ``E. De Castro'', Univ. di Bologna, Italia}
\centerline{e-mail {\tt ferri@dm.unibo.it}}

%%% in case of several authors:
\smallskip
\centerline{Patrizio Frosini}
\centerline{\it ARCES ``E. De Castro'', Univ. di Bologna, Italia}
\centerline{e-mail {\tt frosini@dm.unibo.it}}
%%%

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
\par
%   Text of the abstract
Size functions are modular shape descriptors of a geometrical--topological nature.
Technically, they are invariants of a pair consisting of a manifold and of a continuous
real function (called {\it measuring function}) defined on it. Whatever the input pair,
a size function is always defined on $\mathbb{R}^2$, with values in the extended
integers. Size functions have already been applied to several pattern recognition
and database searching tasks.

The problem of deducing information on the shape of a 3D object from 2D projections
is well--known and is a relevant one. A way to face the problem is obviously to get
several projections and to integrate information as, e.g., in Computer Assisted
Tomography, but this is not always feasible. In computer vision, it is frequent to have
access to just one view of an object.

Size functions can describe shape of objects of any dimension; however, no effort
has been done so long, in order to relate these descriptors applied to a 3D object
and a 2D image of it.

The present research deals with this problem, restricted to the wide and significant
class of convex bodies. Given a bound on the difference between the measuring function
defined in 3D, and the one defined on the projected points, the size function of the 3D
object at a given real pair $(x,y)$ is proved to be bound from both sides by the size
function of the image at nearby pairs. Specific examples are given.


\bigskip

%%% in case of several authors:
{\bf  Presented by Massimo Ferri}
%%%


\pagebreak


\centerline{\large \bf Mixed Support Functions}

%\centerline{\large \bf Title to be Continued Here if Too Long}

\bigskip

\centerline{Paul Goodey}
 \centerline{\it University of Oklahoma, USA}
  \centerline{e-mail {\tt pgoodey@ou.edu}}

%%% in case of several authors:
\smallskip
\centerline{Wolfgang Weil}
 \centerline{\it Universit\"at Karlsruhe, Germany}
  \centerline{e-mail {\tt weil@ma2geo.mathematik.uni-karlsruhe.de}}
%%%

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par
%   Text of the abstract
For convex bodies $K$ and $M$ in $\Bbb E^d$, we will describe a
construction which shows how the $j$-th surface area measure of
$K$ can be combined with the $(d-j+1)$-st surface area measure of
$M$ to form the first surface area measure of a mixed convex body
$T_j(K,M)$. The Minkowski sum of all these mixed bodies, for
$j=1,\dots,d$, is the (centred) sum of all intersections of $K$
with translations of $M$. The latter observation follows from an
integral translative formula of W. Weil which was the motivation
for this study. Our construction relies on a result of R.
Schneider which characterizes first surface area measures of
polytopes.
\bigskip

%%% in case of several authors:
{\bf  Presented by Paul Goodey}
%%%


\pagebreak


  \centerline{\large \bf
  %Title of the Abstract
  Inversion of the Radon transform via G{\aa}rding-Gindikin
  fractional integrals
  }

  \centerline{\large \bf
  %Title to be Continued Here if Too Long
  }

  \bigskip

  \centerline{
  %Author 1
  Eric L. Grinberg
  }
  \centerline{\it
  %Institution Name, Country
  Department of Mathematics, Temple University, Philadelphia PA 19122, USA
  }

  \centerline{e-mail {\tt
  %Author 1
  grinberg@math.temple.edu
  }}

  %%% in case of several authors:
  \smallskip
  \centerline{
  %Author 2
  Boris Rubin
  }
  \centerline{\it
  % Institution Name, Country
  Institute of Mathematics,
  Hebrew University, Jerusalem 91904, ISRAEL
  }
  \centerline{e-mail {\tt
  %Author 2
  boris@math.huji.ac.il
  }}
  %%%

  \bigskip
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  %%
  \par
  %   Text of the abstract
  The Funk-Radon transform has figured prominently in convex geometry
  during the past decade and inversion formulas have played an important
  role. There are many standard inversions, some involving differential
  operators and others involving fractional integrals. Here we consider a
  natural generalization of the Radon transform, which acts on functions
  on Grassmannians to produce functions on other Grassmannians. These
  transforms also occur in convex geometry and have been studied and
  inverted before, e.g., by I.M.~Gelfand and his collaborators. We
  establish a connection between the Radon transform and
  G{\aa}rding-Gindikin fractional integrals associated to the cone of
  positive definite matrices, and use it to obtain Abel type
  representations and inversion formulae on various function spaces,
  including $L^p$.
  \bigskip

  %%% in case of several authors:
  {\bf  Presented by
  %...
  Eric Grinberg}
  %%%


\pagebreak


\centerline{\large \bf Revisited inequalities for convex sets}

\bigskip

\centerline{Stefano Campi}
\centerline{\it
            Dipartimento di Matematica Pura e Applicata ``G.\,Vitali",}
\centerline{\it Universit\`a degli Studi di Modena e Reggio Emilia, ITALIA}
\centerline{e-mail {\tt campi@unimo.it}}


\smallskip
\centerline{Paolo Gronchi}
\centerline{\it Istituto per le Applicazioni del Calcolo,}
\centerline{\it Consiglio Nazionale delle Ricerche, ITALIA}
\centerline{e-mail {\tt paolo@iaga.fi.cnr.it}}
%%%

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par
A significant part of the theory of convex bodies is based on geometric
inequalities. These inequalities are usually expressed in terms of
extrema of affine functionals and from this point of view an important
role is played by ellipsoids and their characterizations.

We present some new applications of a result by Rogers and Shephard on
continuous movements of convex sets. Classical inequalities of Convex
Geometry are rediscovered and connections between open problems are
drafted.

\bigskip


{\bf  Presented by Paolo Gronchi}
%%%


\pagebreak


\centerline{\large \bf Approximating of convex bodies in
  $\mathbb{E}^3$ }

\centerline{\large \bf by circumscribed polytopes}

\bigskip

\centerline{Peter M.~Gruber}
\centerline{\it Technische Universit\"at Wien, Austria}
\centerline{e-mail {\tt peter.gruber@tuwien.ac.at}}

%%% in case of several authors:
%\smallskip
%\centerline{Author 2}
%\centerline{\it Institution Name, Country}
%\centerline{e-mail {\tt Author 2}}
%%%

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par
%   Text of the abstract
In this lecture we describe the asymptotics, as $n\to\infty$, of
the minimum volume difference of a smooth convex body in
$\mathbb{E}^3$ and its circumscribed convex polytopes with $n$
facets. In addition, a rough description of the form of the best
approximating polytopes is given. Some remarks on the
mathematical context of this problem are made.


\bigskip

%%% in case of several authors:
%{\bf  Presented by ...}
%%%


\pagebreak


  \centerline{\large \bf Estimating  the distance to the convex hull}



  \bigskip

  \centerline{S. J. Dilworth}
  \centerline{\it Department of Mathematics, University of South
  Carolina, Columbia, SC 29208,  USA }
  \centerline{e-mail {\tt dilworth@math.sc.edu}}

  %%% in case of several authors:
  \smallskip
  \centerline{Ralph Howard}
  \centerline{\it Department of Mathematics, University of South
  Carolina, Columbia, SC 29208,  USA}
  \centerline{e-mail {\tt howard@math.sc.edu}}
  %%%

  \smallskip
  \centerline{James W. Roberts}
  \centerline{\it Department of Mathematics, University of South
  Carolina, Columbia, SC 29208,  USA}
  \centerline{e-mail {\tt roberts@math.sc.edu}}




  \bigskip
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  %%
  \par
  %   Text of the abstract

  \bigskip

  Let $A$ be a bounded subset of $\mathbf{R}^n$ and let $\operatorname{co}(A)$ be
  the convex hull of $A$.  It is natural to estimate how much larger
  $\operatorname{co}(A)$ is than $A$.  One easily stated  result in this
  direction is\\
  {\bf Theorem} {\em
  Assume that for all $a,b\in A$ that $\operatorname{dist}((a+b)/2,A)\le \delta$.
  Then there is a constant $C(n)$, only depending on the dimension, so
  that}
  $$
       x\in \operatorname{co}(A)\quad    \text{implies}  \quad
  \operatorname{dist}(x,A)  \le C(n)\delta.
  $$
  The sharp constant $C(n)$ is given.  Therefore there is a sharp bound
  for the distance of $A$ to $\operatorname{co}(A)$ in terms of the
  distance from $A$ to the set of all midpoints of segments with
  endpoints in $A$.  Surprisingly, the examples giving the sharp
  constant are fractals. The proof involves a detailed study of
  \emph{almost convex functions}, defined by $ f((x+y)/2) \le
  (f(x)+f(y))/2 + 1$, and how well they can be approximated by convex
  functions.  Several generalizations are given.




  %%% in case of several authors:
  {\bf  Presented by Ralph Howard}
  %%%


\pagebreak



  \centerline{\large \bf The Minkowski Problem for Polytopes}


  \bigskip

  \centerline{Dan Klain}
  \centerline{\it University of Massachusetts Lowell, USA}
  \centerline{e-mail {\tt Daniel\_Klain@uml.edu}}


  \bigskip
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  %%
  \par
  %   Text of the abstract

  The traditional solution to the Minkowski problem for polytopes involves
  two steps.  First, the existence of a polytope satisfying given boundary
  data is demonstrated.  In the second step, the uniqueness of that polytope
  (up to translation) is then shown to follow from the equality conditions
  of Minkowski's inequality, an inequality for mixed volumes that is
  typically proved in a separate context.  In recent work the speaker has
  adapted the classical argument to prove both the existence theorem of
  Minkowski {\em and} his mixed volume inequality simultaneously, thereby
  demonstrating the equiprimordial relationship between these two
  fundamental theorems of modern convex geometry.

  \bigskip



\pagebreak



\centerline{\large \bf The Busemann-Petty problem via spherical harmonics}


\bigskip

\centerline{Alexander Koldobsky}
\centerline{\it University of Missouri-Columbia, USA}
\centerline{e-mail {\tt koldobsk@math.missouri.edu}}



\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
\par
%   Text of the abstract

The Busemann-Petty problem asks whether symmetric
convex bodies in $R^n$ with smaller central hyperplane sections
necessarily have smaller $n$-dimensional volume. The solution has
recently been completed, and the answer is affirmative if $n\le 4$
and negative if $n\ge 5.$ In this article we present a short proof
of the affirmative result and its generalization using the Funk-Hecke
formula for spherical harmonics.

\bigskip




\pagebreak



\centerline{\large \bf Point $X$-rays of convex bodies in planes
of constant curvature}


\bigskip

\centerline{Paolo\ Dulio} \centerline{\it  Politecnico di Milano,
Italy} \centerline{e-mail {\tt paodul@mate.polimi.it}}

\smallskip
\centerline{Carla Peri} \centerline{\it Universit\`a Cattolica,
Italy} \centerline{e-mail {\tt cperi@mi.unicatt.it}}

\bigskip
\par
The $X$-ray\textit{ }of a set $A$ from a point $p$ is the function
giving the total length of the intersection with $A$ of each ray
issuing from $p.$ In a previous paper we showed that $X$-rays from
two points are enough, with some exceptions, to distinguish
between any two convex bodies in non-Euclidean spaces such that
the line joining the two points intersects the interior of the
bodies. Here we prove that any convex body in a plane of constant
curvature is uniquely determined by the $X$-rays from four points
in general positions, so extending results obtained by R. Gardner
and A. Vol\v{c}i\v{c} in Euclidean and projective plane.
\bigskip

{\bf  Presented by Carla Peri}


\pagebreak



\centerline{\large \bf Minkowski sum of quasi-convex functions}

\centerline{\large \bf and p-capacity of convex sets}

\bigskip
\centerline{Andrea Colesanti}
\centerline{\it Dipartimento di Matematica ``U. Dini'', Universit\'a di Firenze, Italy}
\centerline{e-mail {\tt colesant@math.unifi.it}}

%%% in case of several authors:
\smallskip
\centerline{Paolo Salani}
\centerline{\it Dipartimento di Matematica ``U. Dini'', Universit\'a di Firenze, Italy}
\centerline{e-mail {\tt salani@math.unifi.it}}


%%%

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par
%   Text of the abstract
A function $u$ is said {\em quasi-convex} ({\em quasi-concave})
if its sublevel (superlevel) sets are convex.
If $u_0$ and $u_1$ are quasi-convex (quasi-concave) functions, for every
$t\in[0,1]$ we define the Minkowski sum of $u_0$ and $u_1$ as the
function $u_t$ whose sublevel (superlevel) sets are the Minkowski sums of the
corresponding sublevel (superlevel) sets of $u_0$ and $u_1$.

We investigate the relationship between $\nabla u_t$,
$D^2 u_t$ (eventually intended in the viscosity sense) and
$\nabla u_i$, $D^2 u_i$, $i=0,1$, proving that some relevant
differential inequalities satisfied by the original functions
$u_0$ and $u_1$ are inherited by $u_t$.

Applying this to quasi-concave functions in convex rings,
we obtain a new and simple proof of the Brunn-Minkowski inequality
for electrostatic capacity of convex sets and we are able to extend the result to the case of $p$-capacity.
\bigskip

%%% in case of several authors:
{\bf  Presented by Paolo Salani}
%%%


\pagebreak


\centerline{\large \bf Stability results involving surface
area measures }

\centerline{\large \bf of convex bodies}

\bigskip

\centerline{Daniel Hug}
\centerline{\it Universit\"at Freiburg, Germany}
\centerline{hug@email.mathematik.uni-freiburg de }

%%% in case of several authors:
\smallskip
\centerline{Rolf Schneider}
\centerline{\it Universit\"at Freiburg, Germany}
\centerline{rschnei@uni-freiburg.de }
%%%

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
\par
%   Text of the abstract
We report on  stability versions of uniqueness results for convex
bodies in ${\mathbb R}^{d}$, where the uniqueness is implied by
assumptions on area measures of order $d-1$ or lower, or by
assumptions on geometrically significant integral transforms of
such measures. For example, a stability result is obtained for
Lutwak's extension of the Aleksandrov-Fenchel-Jessen theorem to
the Brunn-Minkowski-Firey theory. In a different direction, a
method used by Bourgain and Lindenstrauss in the case of
projection bodies is extended. It yields some stability theorems
for not necessarily symmetric convex bodies.
\bigskip

%%% in case of several authors:
{\bf  Presented by Rolf Schneider}
%%%


\pagebreak



\centerline{\large \bf Affine surface areas and }

\centerline{\large \bf approximation of convex bodies by polytopes}

\bigskip

\centerline {Elisabeth Werner}
\centerline{\it Case Western Reserve University,
USA}
\centerline{e-mail {\tt emw2@po.cwru.edu}}



\bigskip


We discuss the classical affine surface area and its extensions, the
$p$-affine surface areas,
for $- \infty \leq p \leq \infty$. The  $1$-affine surface area is
just the classical affine
surface area.
\par
We show the role it plays in convexity theory in general, and  in particular
in the approximation of convex bodies by polytopes. Various types of
approximation will be considered,
so for instance best approximation and  random approximation.
\par
It is natural that the
affine surface areas should appear in this context as they ``measure"
the boundary
structure of a convex body.
\bigskip



\pagebreak


\centerline{\large \bf Tilings of normed spaces: The state of the
art }


\bigskip

\centerline{Clemente Zanco}
\centerline{\it Universit\`a degli Studi, Milano, Italy}
\centerline{e-mail {\tt zanco@mat.unimi.it}}

\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par
%   Text of the abstract

\bigskip
We illustrate the main results available up to now concerning
tilings of infinite-dimensional normed spaces. In particular, we
discuss the possibility of getting tilings with uniformly bounded
tiles from above and/or from below.
\end{document}

