On some extremal simplexes
Mir Masoom Ali · Pacific Journal of Mathematics · 1970
Let A be a fixed point in ^-dimensional Euclidean space.Let B lf B 2 , '" f B n+ί be the vertices of a simplex S n of ndimensions, that is, the n + 1 vertices do not lie on a (n -1) dimensional subspace.Let d if assumed to be positive, be the distance of Bι from A, and let Uj be the cosine of the angle between the straight lines ABi and ABj for i, j = 1, 2, •••, n + 1.Let πi denote the (n -1 -dimensional hyperplane passing through all the vertices of S n except B if let pi, assumed positive, be the perpendicular distance of πι from A, and let ma denote the cosine of the angle between the normals from A to TΓi and π 3 -for i, j =1,2, , n + 1.The present paper deals with the following problems.(a) An expression for the content of S n , C(S n ) say, in terms of di and Uj for i, j =1,2, --,n + 1 is first obtained.Then leaving d l9 d 2 , -" 9 d n +i fixed, values of l iJ9 say l* 3 , are determined in such a manner that C(S n ) is a maximum, and the maximum value of C(S n ) is obtained for the two cases that arise: (i) when A is inside S n , (ϋ) when A is outside S n .The latter case does not arise when d x = d 2 -= d n +u (b) An expression for C(S n ) is obtained in terms of pi and m>ij, i, j = 1, 2, , n + 1.Then leaving p u p 2 , , p n +i fixed, values for ma, say m%, are determined in such a manner that C(S») is a minimum, and such C(S n ) is computed for the two cases that arise depending on (i) whether A is inside S n or (ii) A is outside S n .The latter case does not arise when Pi = ί>2 = = Pτι + 1 .