On W. Fenchel's solution of the plank problem

We denote by E '' the n-dimensional Euclidean space. A strip S e E '~ is a closed convex set bounded by two parallel hyperplanes. Their distance is the width of S. The vector v is said to be the width vector of S, if S is bounded by hyperplanes normal to v and the width of S is 2Iv I. The width of a convex body K ~ E ~ with inner points is the minimal width of a strip covering K. We denote points also by vectors leading to them from the origin. A. TA•SKI stated in 1932 the following conjecture, known as plank problem: If a convex body K is covered by a finite number of strips, then the sum of their widths is not less than the width of 14. TH. BANG'S [1] proof of this conjecture has been simplified by W. FENCHEL [2]. This simplified proof is based on three simple lemmas, which reduce the conjecture to the following statement: The union of the strips S~,..., Sr of width vectors v~,..., vr does not contain each of the 2" points J~(_+vl+ ... ___v~), where t~>l and the origin is an arbitrary point. Present note gives an alternative proof of this statement. We denote by 1-I the set of the 2 r points / , ( + v ~ + . . . + v , . ) , further by (P,H) and (P,S) the distance of the point P from the hyperplane H and from the strip S, respectively. We start the proof with the special case in which the middle hyperplanes /-/1, . . . . H~ of the strips S, , . . . ,S~ have a common point C. Let P be an element of / / o f maximal distance CP. We show that P is not covered by S~, . . . , S~, i. e. P proves our statement. If the sign of v~ ( i x 1 , . . . , r) is altered in the vector defining P, we get a vector which defines P~ E/7. By definition of P we have C P ~ CP~. > p Consequently, since C~H~ and PPi• we obtain (P,H~)--_-( ,:,H O, (P, H~)>--~PP~=2 N~I and (P, S0>(,~--I)[v~]. This proves that $1 . . . . , S,. contain neither P nor any point Q for which PQ<(i~--I)d, where d min (Iv~I, . . . , ]vrl).