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

$$\begin{aligned} \sigma _{\mathrm{cone}(\mathcal {P})}(\mathbf{z}) = \sum _{j=1}^m \sigma _{\mathbf{v} + \mathcal {K}_j}(\mathbf{z}). \end{aligned}$$

Unraveling this, the series of implications one gets is

$$\begin{aligned} \sum _{j=1}^m \sigma _{\mathbf{v} + \mathcal {K}_j}(\mathbf{z})&= \sum _{j=1}^m \sum _{\mathbf{m} \in (\mathbf{v} + \mathcal {K}_j) \cap \mathbf{Z}^{d+1}} \nu (\mathbf{v} + \mathcal {K}_j, \mathbf{m}) \mathbf{z}^\mathbf{m}\\&= \sum _{\mathbf{m} \in (\mathbf{v} + \mathrm{cone}(\mathcal {P})) \cap \mathbf{Z}^{d+1}} \nu (\mathbf{v} + \mathrm{cone}(\mathcal {P}), \mathbf{m}) \mathbf{z}^\mathbf{m}. \end{aligned}$$

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].