Abstract
For a non-empty compact set A⊂ℝd, 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.
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Mathematics Subject Classification (2000): 52A38, 28A75, 52A22, 53C65
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Heveling, M., Hug, D. & Last, G. Does polynomial parallel volume imply convexity?. Math. Ann. 328, 469–479 (2004). https://doi.org/10.1007/s00208-003-0497-7
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DOI: https://doi.org/10.1007/s00208-003-0497-7