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\topmatter
\specialhead
\centerline{
                            RESEARCH PROBLEM
            }
\endspecialhead
\vskip .4cm

\title
\nofrills{\bigrm 
                 A generalization of the plank problem
          }
\endtitle
\author
Tibor \'Odor
\endauthor
\rightheadtext\nofrills{\sevenrm
                 A generalization of the plank problem
                        }
\leftheadtext\nofrills{\sevenrm Tibor \'Odor}
\address 
Math.\ Inst.\ of the Hung.\ Acad.\ of Sci., P.O.B.\ 127, 
Budapest, H-1364 Hungary
\endaddress
\email 
odor\@math-inst.hu
\endemail
\thanks
Partially supported by Hung.\ Nat.\ Found for Sci.\ Research, number 4427
\endthanks
\date
\today
\enddate
\subjclass
51M16, 52A40, 53C65
\endsubjclass
\keywords
Convex body, width, covering, inequality, plank
\endkeywords
\abstract
Tarski posed the following problem:
if $K$ is a convex body, and $\Cal C$ is a covering of $K$ by the sets
$\Cal C=\{K_i:i=1,\dots,n\}$, then
the sum of the minimal width of the sets $K_i$ is not smaller
than the minimal width of $K$.
This problem was solved by Bang.

Using minimum of the  integrals of the minimal width function,
we generalize the original problem.
\endabstract
\endtopmatter
\document

\define\SOd{\operatorname{{SO_d}}}

Tarski posed the  following famous problem: if $K$ is a convex
body, and $\Cal C$ is a covering of $K$ by the sets $\Cal
C=\{K_i:i=1,\dots,n\}$, then the sum of the minimal width of the sets
$K_i$ is not smaller than the minimal width of $K$.  This problem
was solved by [Bang, 1951].  Our aim is to  generalize this problem.

Let ${\Cal G}_{dk}$ be the manifold of the $k$-planes in the 
$d$-dimensional real vector space $R^d$, where $1\le k\le d-1$.  Let
${\Cal B}({\Cal G}_{dk})$ be the set of Borel sets of  ${\Cal
G}_{dk}$.  Let $\mu:{\Cal B}({\Cal G}_{dk})\to R$ be a fixed positive
probability measure.  If $K$ is a convex body, let $w(K,H_k)$ denote
the $k$-dimensional area of its orthogonal projection to the
$k$-plane $H_k\in {\Cal G}_{dk}$.  Define the $\mu$-width $|K|_\mu$
of the convex body $K$ by the following formula
$$
\inf_{\tau\in \SOd}\int_{{\Cal G}_{dk}} w(K,H_k)\,d\mu_\tau(H_k),
$$
where $\mu_\tau(B)=\mu(\tau^{-1}B)$ for arbitrary Borel set 
$B\in{\Cal B}({\Cal G}_{dk})$
and $\SOd$ is the group of orthogonal linear transformations in $R^d$.

If $K$  is a convex body in the $d$-dimensional Euclidean space, then let
$\Cal C(K)$ be the set of finite coverings of $K$ by convex bodies.
If ${\Cal C}\in {\Cal C}(K)$, then ${\Cal C}=\{K_1,\dots, K_n\}$, and
let $|C|=\sum_{i=1}^j |K_j|$, where $|K_i|$ is the (minimal) width
of the convex body $K_i$. Let
$$
C_{dk}=\inf_{{\Cal C}\in{\Cal C}(K)}\dfrac{|{\Cal C}|}{|K|}.
$$
It is well known that if $2\le k\le d-1$, then $0<C_{dk}<1$, [Bang, 1951], 
[Bang, 1953].

\proclaim{Conjecture 1}
If $K$  is a convex body in the $d$-dimensional Euclidean space, and
$\Cal C=\{K_i:i=1,\dots,n\}$ is a covering of it by convex bodies,
then the sum of the $\mu$-width of the covering is not smaller than
the $\mu$-width of the body $K$, that is
$$
C_{dk}\cdot |K|_\mu\le \sum_{i=1}^n |K_i|\mu.
$$
\endproclaim

If $\mu$ is an atomic measure concentrated in a single point and
$k=1$, then [Bang, 1951] states our  Conjecture 1 as a theorem. 
For $k\ge 2$ (and $\mu$ is concentrated in a single point), 
he states it as a conjecture.  If $\mu$ is the normalized $\SOd$
invariant measure on ${\Cal G}_{dk}$, then Conjecture 1 follows from
the basic properties of the $k$-th outer measure with constant 1,
instead of $C_{dk}$. This statement was stated in [Bang, 1953] without
a detailed proof. 

Assuming that $\mu$ is concentrated in a single point, $k=(d-1)$ and 
$K=B^d$, we will show that our statement holds with constant 1, instead
of $0<C_{d,d-1}\ne 1$. 

Let $\beta(K)$ be the minimum of the breadth function (that is the
minimum of the $(d-1)$-dimensional volume of the projections to hyper
planes  of the body $K$) for the convex body  $K$.
 
\proclaim{Theorem  1}
Let ${\Cal K}$  be a convex covering of the unit ball $B^d$  in the
$d$-dimensional Euclidean  space. Then 
$$
\beta (B^d)\le \sum_{K\in {\Cal K}} \beta (K).
$$
\endproclaim
{\sl Proof.}  
We say that a Borel set  $C$ is cylindrical if there exists  a hyper
plane  $H$ passing through the origin  $O$ and a Borel set  $C_0
\subset H$ such that $C$ is the inverse image of the orthogonal projection
to $H$.

Clearly it is sufficient to prove our statement if the elements
of ${\Cal K}$ are cylindrical sets $C$, for which 
$$
V_{d-1}(C_0)=\beta (C\cap B^d)
\tag{1}
$$ 
holds. Let $\vartheta (r)=(1-r^2)^{-\frac{1}{2}}\pi^{-1}$.
It is clear that for any cylindrical set $C$ for which (1) holds 
it is true that 
$$
\int_{B^d\cap C} \vartheta (|x|)\, dx=
2\cdot \int_{C_0} \int_0^{\sqrt{1-|y|^2}} 
\vartheta (\sqrt{t^2+|y|^2})\, dt \, dy=
$$
$$
2\cdot \int_{C_0} \int_0^{\sqrt{1-|y|^2}} 
\dfrac{1}{\pi \sqrt{1-|y|^2-t^2}}\, dt \, dy=
\dfrac{2}{\pi} \cdot \int_{C_0} \dfrac{\pi}{2}\, dy= 
V_{d-1}(C_0)=\beta (C),
$$
because $\int_0^u\dfrac{1}{\sqrt{u^2-t^2}}\, dt=\dfrac{\pi}{2}$
[Gradshteyn--Ryzhik, 1965].

Using  that $\vartheta$ is positive,  we get  our statement in the
usual way. 
\qued

Define the relative $\mu$-width $|K|_{L,\mu}$ of the convex body $L$
with respect to the convex body $K$ by the following formula:
$$
|K|_{L,\mu}=
\inf_{\tau\in \SOd}\int_{{\Cal G}_{dk}} \dfrac{w(L,H_k)}{w(K,H_k)}
\,d\mu_\tau(H_k).
$$

A stronger version of Conjecture 1 is as follows.
\proclaim{Conjecture 2}
If $K$ is a convex body in the $d$-dimensional Euclidean space, and
$\Cal C=\{K_i:i=1,\dots,n\}$ is a covering of it by convex bodies,
then the sum of the relative $\mu$-widths with respect to the body $K$
of the covering is not smaller than $C_{dk}$, that is
$$
C_{dk}\le \sum_{i=1}^n |K_i|_{L,\mu}
$$
\endproclaim
This statement is not proved even in the classical case, when the
measure $\mu$ is concentrated in a single point and $k=1$.

\heading
Acknowledgements
\endheading
I am indebted to Endre Makai, Tam\'as Hausel, G\'abor Fejes T\'oth
and L\'aszl\'o Szab\'o for the valuable information and suggestion.


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\endRefs


\enddocument
