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.