A treatise of Archimedes: Geometrical solutions derived from by Heiberg J.L. (ed.)

By Heiberg J.L. (ed.)

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Fields p e r p e n d i c u l a r t o t h e h a l f l i n e s from t h e o r i g i n , i . e . 1 These s u r f a c e s w e s h a l l c a l l c o n e s . A D I F F E R E N T I A L INEQUALITY FOR MINIMAL CONES. "If THM. if a s vertex. t. Y x o 1Xl2 A s usual c2 = L. ( ~ . V, . )=v ~ i ,7 PROOF. c 1 1 From t h e d e f i n i t i o n o f Substituting 6,6, with GiSh+ and c2 and A A = LAhAh. h , we have E ( v ~ ~ v ~) 6 kv, ~ - v . ~ i h k we obtain k ,6ivj6h6h6ivj= = h,i,i 6ivj6h6i6hvj = h,i,j -L 6hvi6hvk6ivj6kvj h,k,i,j I MINIMAL CONES x6hVh f o r which we have used t h e i d e n t i t i e s h EVh6h = 0 h i n p l a c e of (vh6ivk-vi6hvk)~k Writing again 23 = 0 , 6,6, .

A f t e r some p r e l i m i n a r i e s a b o u t Radon measures, w e s h a l l d e f i n e t h e p e r i m e t e r of a Lebesgue measurable s e t and p r o v e i t s g l o b a l and l o c a l properties. 1 W e s h a l l p r e s e n t i n t h i s s e c t i o n some b a s i c f a c t s a b o u t Radon measures and e x t e r i o r measures i n an e u c l i d e a n s p a c e . Particularly for the differ- e n t i a t i o n of measures w e f o l l o w c l o s e l y t h e " G e o m e t r i c Measure Theory" t e x t by H.

1 T o take care of the condition lKj B(U B ) < h h is the sequence of cubes of E n + l j (s = 1 ,Z,. UhLjCh is closed and contained We may then conclude +m , let us observe that if cut by the hyperplanes x = z we obviously have U . )1 7 < +a , then 49 SETS OF F I N I T E PERIMETER , V BE proved t h a t L e t u s p u t now ct = (31 c1 the extension of I V E>0 A to ; . {(UhBh)ll K . } E 3 * I t i s now e a s i l y s e e n t h a t I g e b r a c o n t a i n i n g all bounded open s e t s , hence , 3 ADB f3* open and C B*cB Then = $ .

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