THE CLOSEST PACKING OF CONVEX TWO-DIMENSIONAL DOMAINS, CORRIGENDUM
BY C. A. R O G E R S
London
S o m e t i m e ago I p u b l i s h e d a n a c c o u n t ( I ) , in outline, of c e r t a i n r e s u l t s , w h i c h h a d b e e n a n t i c i p a t e d a n d l a r g e l y s u p e r s e d e d b y w o r k of L. F e j e s T6th(2). I n o w f i n d t h a t i t is n e c e s s a r y t o c o r r e c t one of t h e results.
L e t K be a n o p e n c o n v e x t w o - d i m e n s i o n a l set. A s y s t e m K + a 1, K + a s , ... of t r a n s l a t e s of K b y v e c t o r s el, as, ... is called a p a c k i n g , if no t w o of t h e sets h a v e a n y p o i n t in c o m m o n . L e t d (K) d e n o t e t h e lower b o u n d of t h e d e t e r m i n a n t s of t h e l a t t i c e s A , w i t h t h e p r o p e r t y t h a t t h e s y s t e m of t r a n s l a t e s of K b y t h e v e c t o r s of A f o r m s a p a c k i n g .
M y 1951 p a p e r o n l y p r o v e s
T ~ ] ~ o ~ ] ~ l a . Let K and S be a n y open bounded convex sets with areas a ( K ) and a ( S ) . Let K be symmetrical. I / n sets K can be packed into S (with n>~ 1), then
(n - 1) d (K) + a (K) ~< a (S).
I t i n c o r r e c t l y claims t o p r o v e such a r e s u l t w i t h o u t t h e s u p p o s i t i o n t h a t K s h o u l d be s y m m e t r i c a l . N o r e s t r i c t i o n to s y m m e t r i c a l sets is n e e d e d i n T h e o r e m 2, n o r in t h e m a i n conclusion t h a t i t is i m p o s s i b l e t o f i n d a p a c k i n g of s i m i l a r l y o r i e n t a t e d c o n g r u e n t c o n v e x d o m a i n s , w h i c h is closer t h a n t h e closest l a t t i c e p a c k i n g of t h e d o m a i n s .
T h e e r r o r arises in t h e p r o o f of L e m m a 5; t h e r e is no j u s t i f i c a t i o n for t h e asser- t i o n t h a t i t is p e r m i s s i b l e t o s u p p e s e t h a t t h e p o i n t 1 (c + d) coincides w i t h t h e origin, since in t h i s l e m m a a c h a n g e of origin changes t h e a r e a of t h e p o l y g o n Y[. I t is e a s y to see t h a t t h i s m o v e m e n t of t h e origin i n c r e a s e s t h e a r e a of I I b y 89 I b - a I" (h~ - hi),
(1) Acta Mathematica, 86 (1951), 309-321.
(3) Acta Sci. Math. (Szeged), 12 (1950), 62-67.
306 c . A . ROCERS
where t b - a l is t h e l e n g t h of the segment a b a n d hi, h 2 are the perpendieu]ar dis- tances from t h e origin, supposed in K, to t h e tae-lines to K parallel to the line a b.
Consequently, in L e m m a 5, t h e inequality (8) needs t o be replaced b y
( m + i n - l ) d ( g ) 4 a ( I I ) + 89 ] b - a I ( h 2 - h l ) . (8a) I n order to state t h e a p p r o p r i a t e l y revised version of T h e o r e m 1, it is convenient to i n t r o d u c e the mixed area of two convex domains. F o r our purpose, it is sufficient to define the mixed area a (P, K) of two convex domains P a n d K, b y writing
a ( P + K ) = a ( P ) § 2 a ( P , K ) + a ( K ) ,
where P + K denotes t h e v e c t o r s u m of the sets P a n d K. We indicate how t h e following result m a y be established.
THEOREM l b . Let K and P be a n y open bounded convex sets. I / a 1 .. . . . an are n points o/ P (with n>~ 1), and the sets K § a 1 . . . K + an /orm a packing, then
(n - 1) d (K) ~< a (P) + a (P, K) + a (P, - K). (2 b) N o t e t h a t when K is s y m m e t r i c a l this i n e q u a l i t y reduces to t h a t of T h e o r e m 1 a.
If Q is a n y convex set strictly included in P , we have
a ( Q ) + a ( Q , K ) + a ( Q , - K ) < a ( P ) + a ( P , K ) + a ( P , - K ) ;
it follows, essentially as in t h e ' p r o o f ' of T h e o r e m 1 given before, t h a t we m a y sup- pose t h a t P is minimal. As before this implies t h a t the sets K + a~, K + a 2 , ..., K + a,~
m u s t t o u c h each other in t h e w a y described. This enables us to a p p l y t h e corrected f o r m of L e m m a 5 a n d to o b t a i n the inequality (2 b) w i t h o u t difficulty.
R e c e n t l y N. Oler (1) a n d H. L. Davies (2) have o b t a i n e d results on t h e r a c k i n g of convex s y m m e t I i c a l domains which are significantly b e t t e r t h a n T h e o r e m 1 a. While it should be pcssible to use these results to o b t a i n an i m p r o v e m e n t of T h e o r e m 1 b, it does n o t seem to be easy to combine elegance with refinement.
Received August 18, 1960
(1) To appear in Acta Mathematica, vol. 105.
(2) Personal communication.
