Avoid common mistakes on your manuscript.
1 Correction to: Beitr Algebra Geom https://doi.org/10.1007/s13366-022-00621-7
The proof of the original article, Theorem 2.6 contains an error. However all statements in the article remain correct. To be precise, the mistake lies in the equation that in the paper is wrongly given as
Instead of the multiplying factor \(\frac{1}{d!}\), the summands of \(\rho _{IP}(u)\) should be multiplied by \(\frac{1}{(d-1)!}\). Therefore the correct version is
The same happens in the proofs of Theorem 3.2 and Proposition 5.5. Similarly, in line 13 of the original article, Algorithm 1 and line 5 of the original article, Algorithm 2, formula
As a consequence, the examples in the paper are also affected by this mistake. We list here the examples and provide the correct expressions. We note that the figures remain correct.
Example 2.3
If the intersection is a square, then the radial function in a neighborhood of that point will be a constant term over a coordinate variable, e.g. \(\frac{4}{z}\). On the other hand, when the intersection is a hexagon, the radial function is a degree two polynomial over xyz.
Example 2.8
In six regions the radial function has the following shape (up to permutation of the coordinates and sign):
There are then \(18 = 6 + 12\) regions in which the radial function looks like
In the remaining six regions we have
Example 5.3
The facet exposed by the vector (1, 0, 0) is the intersection of \(z = 4\) with the convex cone
In other words, the variety \({\mathcal {V}}(z-4)\) is one of the irreducible components of \(\partial _a IP\). The remaining 8 regions are spanned by 3 rays each, and the polynomial that defines the boundary of IP is a cubic, such as
in the region
Example 5.8
The polynomial that defines the boundary of IP in the region \({\overline{C}}_1\) is a quartic, namely
The polynomial that defines the boundary in the region \({\overline{C}}_2\) is a cubic
Example 5.10
In the 12 regions which are spanned by five rays, the polynomial that defines the boundary of IP has degree 5 and it looks like
In the other 20 regions spanned by three rays, \(\partial IP\) is the zero set of a sextic polynomial with the following shape
Example 6.2
The linear face exposed by (1, 0, 0, 0) is cut out by the hyperplane \( w = 8\). The second family of chambers is made of cones with 5 extreme rays, where the boundary is defined by a cubic equation with shape
Finally there are 64 cones spanned by 4 rays such that the boundary of the intersection body is a quartic, such as
References
Berlow, Katalin, Brandenburg, Marie-Charlotte., Meroni, Chiara, Shankar, Isabelle: Intersection bodies of polytopes. Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 63(2022),(January 2022)
Open Access
This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Berlow, K., Brandenburg, MC., Meroni, C. et al. Correction to: Intersection bodies of polytopes. Beitr Algebra Geom 63, 441–443 (2022). https://doi.org/10.1007/s13366-022-00638-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13366-022-00638-y