Let A be a fixed point in
ndimensional Euclidean space. Let B_{1},B_{2},⋯,B_{n+1} 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_{i}, assumed to be positive, be the distance of B_{i} from A, and let l_{ij} be
the cosine of the angle between the straight lines AB_{i} and AB_{j} for i,j = 1,2,⋯ ,
n + 1. Let π_{i} denote the (n − 1)dimensional hyperplane passing through all the
vertices of S_{n} except B_{i}, let p_{i}, assumed positive, be the perpendicular distance of π_{i}
from A, and let m_{ij} denote the cosine of the angle between the normals from A to π_{i}
and π_{j} 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 d_{i} and l_{ij} for
i,j = 1,2,⋯,n + 1 is first obtained. Then leaving d_{1},d_{2},⋯,d_{n+1} fixed, values of l_{ij},
say l_{ij}^{∗}, 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}, (ii) when A is outside S_{n}. The latter case does not arise when
d_{1} = d_{2} = ⋯ = d_{n+1}.
(b) An expression for C(S_{n}) is obtained in terms of p_{i} and m_{ij},i,j = 1,2,⋯,n + 1.
Then leaving p_{1},p_{2},⋯,p_{n+1} fixed, values for m_{ij}, say m_{ij}^{∗}, are determined in such a
manner that C(S_{n}) 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
The results are stated below.
(a) The content of S_{n},maxC(S_{n}) and l_{ij}^{∗} are given by
 (1.1) 
 (1.2) 
 (1.3) 
where u satisfies the equation
 (1.4) 
The unique negative root for u in (1.4) corresponds to the case when A is inside S_{n}.
When the relation
is not satisfied, the smallest positive root for u in (1.4) corresponds to the case when
A is outside S_{n}. Other roots for u in (1.4), if any, are inadmissible.
(b) The content iC(S_{n}), min(C(S_{n})) and m_{ij}^{∗} are given by
where M_{ii} is the cof actor of m_{ii} in (m_{ij}) and
and
where v satisfies the equation
The unique negative root for v in (1.8) corresponds to the case when A is inside
S_{n}. When the relation
is not satisfied, the smallest positive root for v in (1.8) corresponds to the case when
A is outside S_{n}. All other roots, if any, are inadmissible.
When d_{1} = d_{2} = ⋯ = d_{n+1}, we obtain the special result that the largest
simplex inscribed in a sphere of ndimensions is a regular one, while when
p_{1} = p_{2} = ⋯ = p_{n+1} the smallest simplex circumscribing a sphere is a regular
one.
