Abstract
Cauchy's surface area formula expresses the surface area of ad-dimensional convex body in terms of the mean value of the volume of its orthogonal projections onto (d−1)-dimensional linear subspaces. We consider here averages of the same kind as those in Cauchy's formula but with respect to some direction dependent density function and investigate the stability problem whether the density must be close to 1 if the formula produces approximately the correct surface area. It will be shown that this relationship between surface area and density is, in general, unstable; but if the density function satisfies suitable regularity conditions, then explicit stability estimates can be obtained.
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