The area function of a
convex body K in Euclidean n-space is a particular measure over the field
ℬ 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 ℬ. To within a binomial coefficient, the coefficient of
tn−p−1 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 ℬ 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 ℬ. 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.
