By H. H. Schaefer
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Extra resources for Banach Lattices and Positive Operators (Grundlehren Der Mathematischen Wissenschaften Series, Vol 215)
Such general polar correspondences are studied in projective geometry. In our case we treat the polar correspondence with respect to the unit ball. 5 The Spherical Image 47 corresponding vertex Ai of the polygon Q1 lies on the ray ei starting at O1 parallel to ei (since the plane E intersects the planes R and R1 in parallel straight lines). If we take the projection of the polygon Q1 to the plane R of the polygon Q, then by the above we obtain a polygon Q whose vertices lie on rays perpendicular to the sides of the polygon Q.
This plane intersects P in a convex polygon whose sides lie on the surface of P and consequently belong to the convex hull R. However, if all edges of a convex polyhedron lie in some convex ﬁgure R, then the polyhedron itself is contained in the ﬁgure. , P is contained in R. Thus, we have proved that both P contains R and R contains P ; hence, the two sets coincide. Therefore, their boundaries coincide, implying that the polyhedron P is the boundary of the convex hull R of its vertices and the limit angle, which is the required conclusion.
Taking the vertices of P as these points, we see that R also includes all unbounded edges of R. Thus, R includes all edges of P . However, if a convex ﬁgure includes all edges of a convex polygon, then this ﬁgure obviously includes the polygon itself. (To prove this, let Q be a convex polygon and let A be one of its vertices (Fig. 23). The polygon Q is contained in the angle between the half-lines p and q starting at A along the adjacent edges. Let X be an arbitrary point of Q. Draw a straight line through X intersecting p and q.
Banach Lattices and Positive Operators (Grundlehren Der Mathematischen Wissenschaften Series, Vol 215) by H. H. Schaefer