Let C be a compact set in Rn.
The r-parallel body of C, Br(C), is the union of the family of closed r-balls whose
centers lie in C. If C is starshaped with respect to the origin, the gauge of Br(C) is a
Lipschitz function; this observation in conjunction with the Arzela-Ascoli theorem
yields Blaschke selection theorem for starshaped sets. In addition, each parallel body
is a union of a finite collection of parallel bodies of starshaped sets. From this
decomposition, we show that Lebesgue measure is continuous on the metric
space of parallel bodies of a fixed radius in Rn relative to the Hausdorff
metric.
