The proofs of [BRS, Theorems 3, 4] given in [BRS, §6] contain an error, and indeed, the statements are not correct in the level of generality given there. We thank Katharina Jochemko and Raman Sanyal for pointing out the mistake to us. To be precise, the mistake lies in the assertion that equation (9) implies that
Unraveling this, the series of implications one gets is
We had implicitly replaced \(\nu (\mathbf{v} + \mathrm{cone}(\mathcal {P}), \mathbf{m})\) with \(\nu (\mathbf{v} + \mathrm{cone}(\mathcal {P}), \mathbf{v} + \mathbf{m})\) and used translation-invariance to replace the second quantity with \(\nu (\mathrm{cone}(\mathcal {P}), \mathbf{m})\). However, \(\nu (\mathbf{v} + \mathrm{cone}(\mathcal {P}), \mathbf{m})\) and \(\nu (\mathbf{v} + \mathrm{cone}(\mathcal {P}), \mathbf{v} + \mathbf{m})\) are in general distinct without further properties on \(\nu \).
We refer to [JS] for a corrected version of those theorems as well as counterexamples.
Finally, the valuation \(\nu '(K, \mathbf{x}) = \#(K \cap \{\mathbf{x}\} \cap \mathbf{Z}^d)\) is not translation-invariant, as was stated in the introduction of [BRS].
References
Beck, M., Robins, S., Sam, S.V.: Positivity theorems for solid-angle polynomials. Beitr. Algebra Geom. 51(2), 493–507 (2010). arXiv:0906.4031v2
Jochemko, K., Sanyal, R.: Combinatorial positivity of translation-invariant valuations (2015). arXiv:1505.07440v1
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Beck, M., Robins, S. & Sam, S.V. Corrigendum to: Positivity theorems for solid-angle polynomials. Beitr Algebra Geom 56, 775–776 (2015). https://doi.org/10.1007/s13366-015-0259-1
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DOI: https://doi.org/10.1007/s13366-015-0259-1