Beitr\"age zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 35 (1994), No. 1, 131-139.
On Affine Subspaces
K\'aroly Bezdek
Abstract.
Let ${\cal K}$ be a convex body in $E^d$ and let
$0 \leq l \leq d-1$. Then let $I_l({\cal K})$ be the smallest
number of affine subspaces of dimension $l$ lying in $E^d\setminus {\cal K}$
that illuminate ${\cal K}$. We give three equivalent definitions of
$I_l({\cal K})$ extending an earlier result of Boltjanskii and Soltan.
The main result of the paper is that any convex body in $E^d$,
$d\ge 3$ can be illuminated by two $(d-2)$--dimensional affine
subspaces. In
particular, any $3$--dimensional convex body can be illuminated by
two lines in $E^3$. Also, we give a class of the convex bodies in
$E^d$ that can be illuminated by two
$\lfloor \frac d2\rfloor$--dimensional affine subspaces.