Vol. 35, No. 2, 1970

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Local behaviour of area functions of convex bodies

William James Firey

Vol. 35 (1970), No. 2, 345–357

The area function of a convex body K in Euclidean n-space is a particular measure over the field of Borel sets of the unit spherical surface. The value of such a function at a Borel set ω is the area of that part of the boundary of K touched by support planes whose outer normal directions fall in ω. In particular the area function of the vector sum K + tE, where t is nonnegative and E is the unit ball, is a polynomial of degree n 1 in t whose coefficients are also measures over . To within a binomial coefficient, the coefficient of tnp1 in this polynomial is called the area function of order p. For p = 1 and p = n 1 necessary and sufficient conditions for a measure over to be an area function of order p are known, but for intermediate values of p only certain necessary conditions are known. Here a new necessary condition is established. It is a bound on those functional values of an area function of order p which correspond to special sets of . These special sets are closed, small circles of geodesic radius α less than π∕2; the bound depends on α,p and the diameter of K. This necessary condition amplifies an old observation: area functions of order less than n 1 vanish at Borel sets consisting of single points.

Mathematical Subject Classification
Primary: 28.80
Secondary: 52.00
Received: 6 June 1968
Revised: 30 December 1969
Published: 1 November 1970
William James Firey