Beitraege zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 34 (1993), No. 2, 163-171.}

Perturbations and Approximations of Support Functions of Convex Bodies

H. Groemer

Abstract.

Let $h$ be the support function of a
$d$-dimensional convex body and $\Phi$ a real valued function on
the unit sphere. The primary objective of this note is to find
conditions on $\Phi$ that imply that $h+\Phi$ is again a support
function. It is shown that for a large class of convex bodies such
a condition can be formulated as an inequality involving $\Phi$ and
its first and second order partial derivatives. Furthermore, some
applications regarding the approximation of support functions by
sums of spherical harmonics are discussed.