Does polynomial parallel volume imply convexity?

Abstract

Abstract. For a non-empty compact set A ⊂ R d , d ≥ 2, and r ≥ 0, let A ⊕r denote the set of points whose distance from A is r at the most. It is well-known that the volume, V d (A ⊕r ), of A ⊕r is a polynomial of degree d in the parameter r if A is convex. We pursue the reverse question and ask whether A is necessarily convex if V d (A ⊕r ) is a polynomial in r. An affirmative answer is given in dimension d = 2, counterexamples are provided for d ≥ 3. A positive resolution of the question in all dimensions is obtained if the assumption of a polynomial parallel volume is strengthened to the validity of a (polynomial) local Steiner formula. (2000): 52A38, 28A75, 52A22, 53C65 Mathematics Subject Classification

Polynomial parallel volume, convexity and contact distributions of random sets

“…Note that in the two-dimensional special case of this result, which was first established in [8], the integration over the rotation group has no effect. Finally, Section 2 contains some geometric preparations which are needed for the proofs of our main results.…”

“…The aim of this subsection is to establish the following generalization of Theorem 1 in [8]. The result will be extended to higher dimensions and to random sets subsequently.…”

The parallel volume at large distances

“…Heveling, Hug and Last [2] proved that a body in the Euclidean plane is convex, iff its parallel volume is a polynomial. This result was generalized by Hug, Last and Weil [3] to more general 2-dimensional Minkowski spaces and to random sets.…”

“…This leads to a new proof of the following theorem by Heveling, Hug and Last [2]: Theorem 1.2. Let K ⊆ R 2 be a body, for which…”

The derivative of the parallel volume difference

“…Kneser [11] and Sz.-Nagy [18] obtained inequalities saying that the parallel volume of a fixed body considered as function of the distance cannot be "too convex". Heveling, Hug and Last [5] showed that a planar body can only have polynomial parallel volume if it is convex (see also [7]).…”