# Download e-book for kindle: Convexity (Proceedings of symposia in pure mathematics, by Victor L. (editor) Klee

By Victor L. (editor) Klee

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Additional info for Convexity (Proceedings of symposia in pure mathematics, Vol.7)

Sample text

Take of the above neighbourand let E1 be the set of all singular hoods a neighbourhood N1 of points of S to which correspond pairs belonging to N1. If A', A" are an arbitrary pair of points of E1 and (P', Q'), (P", Q") the corresponding pairs of N1 so that A' is on the edge P'Q' and in the interior of (P", Q") and A" is on the edge of P"Q" and in the interior (P', Q') then the angle between lie on a rectifiable curve and P,Q, and A'A" is

Q')} < 8, >0, is the 8-neighborhood of (P, Q). LEMMA 1. Given a Pair (P, Q) with angle >0, and e > 0 there exists 8 > 0 such that for any two pairs (P', Q') and (P", Q") of the 8-neighbourhood of (P, Q) (1) The angle between the edges P'Q' and P"Q" is

THEOREM The set (S } is of a-finite linear measure. Given two closed convex curves C', C" in the two different parallel planes Z = Z', Z = Z". h. C' U C") and by 2" the set of all vectors lying in 2' which meet each of the curves C' and C". We shall first prove - LEMMA. AZ' is finite. Consider a variable plane P touching C' and C" and having them on the same side. If p touches C' and C" each at a single point then the segment joining them lies on I. If p touches one of the curves in a point and the other one in a segment then the triangle with the point and the segment as a vertex and a base respectively lies in 2'.