Vol. 33, No. 1, 1970

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ISSN: 0030-8730
On some extremal simplexes

Mir Maswood Ali

Vol. 33 (1970), No. 1, 1–14
Abstract

Let A be a fixed point in n-dimensional Euclidean space. Let B1,B2,,Bn+1 be the vertices of a simplex Sn of n-dimensions, that is, the n + 1 vertices do not lie on a (n 1) dimensional subspace. Let di, assumed to be positive, be the distance of Bi from A, and let lij 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 Sn except Bi, let pi, assumed positive, be the perpendicular distance of πi from A, and let mij denote the cosine of the angle between the normals from A to πi and πj for i,j = 1,2,,n + 1. The present paper deals with the following problems.

(a) An expression for the content of Sn,C(Sn) say, in terms of di and lij for i,j = 1,2,,n + 1 is first obtained. Then leaving d1,d2,,dn+1 fixed, values of lij, say lij, are determined in such a manner that C(Sn) is a maximum, and the maximum value of C(Sn) is obtained for the two cases that arise: (i) when A is inside Sn, (ii) when A is outside Sn. The latter case does not arise when d1 = d2 = = dn+1.

(b) An expression for C(Sn) is obtained in terms of pi and mij,i,j = 1,2,,n + 1. Then leaving p1,p2,,pn+1 fixed, values for mij, say mij, are determined in such a manner that C(Sn) is a minimum, and such C(Sn) is computed for the two cases that arise depending on (i) whether A is inside Sn or (ii) A is outside Sn. The latter case does not arise when

p1 = p2 = ⋅⋅⋅ = pn+1.

The results are stated below.

(a) The content of Sn,maxC(Sn) and lij are given by

n!C(Sn) = |(lijdidj +1)|1∕2
(1.1)

            2     −1n∏+1  2
max(n!C(Sn)) = − u     (di − u)
i=1
(1.2)

 ∗
lij = u∕(didj) for i,j = 1,2,⋅⋅⋅ ,n + 1; i ⁄= j,
(1.3)

where u satisfies the equation

     n∑+2  2    −1
1 + u   (di − u) = 0.
i=1
(1.4)

The unique negative root for u in (1.4) corresponds to the case when A is inside Sn. When the relation

d1 = d2 = ⋅⋅⋅ = dn+1

is not satisfied, the smallest positive root for u in (1.4) corresponds to the case when A is outside Sn. Other roots for u in (1.4), if any, are inadmissible.

(b) The content iC(Sn), min(C(Sn)) and mij are given by

                          n∏+1
(n!C(Sn))2 = |(pipj + mij)|n∕  |Mii|             (1.5)
i=1

where |Mii| is the cof actor of mii in |(mij)| and

           2     −1 2nn∏+1  2
min(n!C (Sn )) = − v  n     (pi − v)            (1.6)
i=1

and

m∗ = v∕(p p) for i ⁄= j; i,j = 1,2,⋅⋅⋅ ,n+ 1;      (1.7)
ij      ij

where v satisfies the equation

    n∑+1
1+ v   (p2i − v)−1 = 0.                 (1.8)
i=1

The unique negative root for v in (1.8) corresponds to the case when A is inside Sn. When the relation

pi = p2 = ⋅⋅⋅ = pn+1

is not satisfied, the smallest positive root for v in (1.8) corresponds to the case when A is outside Sn. All other roots, if any, are inadmissible.

When d1 = d2 = = dn+1, we obtain the special result that the largest simplex inscribed in a sphere of n-dimensions is a regular one, while when p1 = p2 = = pn+1 the smallest simplex circumscribing a sphere is a regular one.

Mathematical Subject Classification
Primary: 52.40
Milestones
Received: 23 July 1969
Published: 1 April 1970
Authors
Mir Maswood Ali