Dr Burnside, Convex Solids in Higher Space 437 



Convex Solids in Higher Space. By Dr W. Burnside, Honorary 

 Fellow of Pembroke College. 



[Received 14 July 1921.] 



Definitions. A set of linear (n- l)-spreads in w-dimensional 

 space is said to be "general" when no n+ 1 of them meet in a 



point, no w in a hne, no n — 1 in a 2-spread, , and no three 



in an {n— 2)-spread. 



A set of points in ^-dimensional space are said to be the internal 

 points of a convex polyhedron, when each pair A and B of them 

 satisfy the following conditions : (i) no point of the finite hne AB 

 hes on any one of a certain set of m (n — l)-spreads; (ii) the hne 

 AB produced from B meets at least one of the m {n— l)-spreads 

 at a finite distance; (iii) the hne BA produced from A meets at 

 least one of the m (n— l)-spreads at a finite distance. If, in addi- 

 tion, it is always possible to choose B so that AB produced from B 

 meets any assigned one of the m {n- l)-spreads before it meets 

 any of the others, then each of the m {n- l)-spreads is said to 

 form part of the boundary of the convex polyhedron. 



Consider five 3-spreads A, B, C, D, E in 4-dimensional space 

 of "general" position and such that no one of their five points of 

 intersection is at infinity. Denote by a, b, c, d, e the points of 



intersection oi B, C, D and E; ; A, B, C and D. If e and q are 



on opposite sides of E, then eq produced from q does not meet 

 A, B, C, D or E; and therefore q cannot be an internal point of 

 a convex polyhedron bounded by the five 3-spreads. If e and q 

 are on the same side of E, let eq meet E in f. In E the four points 

 a, b, c, d are the vertices of a tetrahedron. If p is outside this 

 tetrahedron, it must be separated from one of the vertices, say a, 

 by the plane through the other three. Hence p and a, and therefore 

 also q and a, are on opposite sides of A : and q cannot be an internal 

 point of a convex polyhedron bounded by the five 3-spreads. It 

 follows that the only points that can be internal points of a convex 

 polyhedron bounded by the five 3-spreads are the points of the 

 finite fines joining e to every internal point of the tetrahedron 

 abed : and these points clearly satisfy all the conditions. 



Hence five 3-spreads of general position in 4-dimensional space, 

 whose intersections are all finite points, form the boundary of just 

 one convex polyhedron. An obvious extension of this reasoning 

 shows that n+l{n— l)-spreads in n-dimensional space, of general 

 position, no one of whose intersections is at infinity, form the 

 boundary of just one convex polyhedron. 



