\documentstyle{amsppt}
\redefine\frac{\tfrac}

\topmatter
\title Planar sections of convex bodies
\endtitle
\author V. V. Makeev
\endauthor
\endtopmatter
\define\Vol{\operatorname{Vol}}
\define\INT{\operatorname{int}}

\subheading{Notation}
Let $X\subset\Bbb R^3$ be a centrally symmetric planar convex set
with center $P$, and let $a\in \Bbb R$.
Then $L(X)$ denotes the affine hull (the plane) of $X$,
and $aX\subset L(X)$ is the homothetic image of $X$ with coefficient $a$
and center $P$.

Let $K \subset \Bbb R^3$ be a convex body.
We denote by $\INT K$ the interior of $K$.

\proclaim{Theorem 1}
$K $ contains
an affine image $C$ of a circular cylinder
with
$$
\Vol (C) \ge \frac{4\pi}{27\sqrt 3}\, \Vol (K).
$$
The estimate is sharp for tetrahedra.
\endproclaim

\proclaim{Theorem 2}
Let $O \in \INT K$.
Then $K$ is circumscribed about an affine-regular octagon $A$
such that $O\in L(A)$.

\endproclaim

\proclaim{Corollary}
Let $O \in \INT K$.

\roster
\item
There is an ellipse $E$ such that $O\in L(E)$
and
$$
E \subset K\cap L(E) \subset \sqrt2 \,E.
$$
The estimate $\sqrt2$ is sharp for tetrahedra.

\item
There is an affine-regular hexagon $H$
such that $O\in L(H)$ and
$$
H \subset K\cap L(H) \subset \frac{5+4\sqrt 2}7\, H.
$$
\endroster
\endproclaim


\proclaim{Theorem 3}
Let $K$ be centrally-symmetric with center $O\in \Bbb R^3$.

\roster
\item
There an is ellipse $E$ with center $O$
such that
$$
E \subset K\cap L(E) \subset \frac2{\sqrt3}\,E.
$$
The estimate $2/{\sqrt3}$ is sharp
for parallelepipeds.

\item
There is
an affine-regular mirror symmetric hexagon $H_1$
with center $O$
such that
$$
H_1 \subset K \cap L(H_1) \subset \frac{2+ \sqrt3}3\, H_1.
$$

\item
There is a regular hexagon $H_2 $ with center $O$
such that
$$
H_2 \subset K \cap L(H_2) \subset \frac32\, H_2.
$$ % 1.5?
The estimate $3/2$ is sharp
for parallelepipeds with one edge much longer than the others.
\endroster

\endproclaim

\end