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The hyperplanes dissect $$\\mathbb R^d$$<\/jats:tex-math>\n \n R<\/mml:mi>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> into finitely many polyhedral chambers. For a point $$x\\in \\mathbb R^d$$<\/jats:tex-math>\n \n x<\/mml:mi>\n \u2208<\/mml:mo>\n \n R<\/mml:mi>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and a chamber P<\/jats:italic> the metric projection of x<\/jats:italic> onto P<\/jats:italic> is the unique point $$y\\in P$$<\/jats:tex-math>\n \n y<\/mml:mi>\n \u2208<\/mml:mo>\n P<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> minimizing the Euclidean distance to x<\/jats:italic>. The metric projection is contained in the relative interior of a uniquely defined face of P<\/jats:italic> whose dimension is denoted by $$\\text {dim}(x,P)$$<\/jats:tex-math>\n \n dim<\/mml:mtext>\n (<\/mml:mo>\n x<\/mml:mi>\n ,<\/mml:mo>\n P<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We prove that for every given $$k\\in \\{0,\\ldots , d\\}$$<\/jats:tex-math>\n \n k<\/mml:mi>\n \u2208<\/mml:mo>\n {<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n ,<\/mml:mo>\n d<\/mml:mi>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the number of chambers P<\/jats:italic> for which $$\\text {dim}(x,P) = k$$<\/jats:tex-math>\n \n dim<\/mml:mtext>\n (<\/mml:mo>\n x<\/mml:mi>\n ,<\/mml:mo>\n P<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> does not depend on the choice of x<\/jats:italic>, with an exception of some Lebesgue null set. Moreover, this number is equal to the absolute value of the k<\/jats:italic>-th coefficient of the characteristic polynomial of the hyperplane arrangement. In a special case of reflection arrangements, this proves a conjecture of Drton and Klivans [A geometric interpretation of the characteristic polynomial of reflection arrangements. Proc. Amer. Math. Soc. 138<\/jats:bold>(8), 2873\u20132887 (2010)].<\/jats:p>","DOI":"10.1007\/s00454-023-00577-y","type":"journal-article","created":{"date-parts":[[2023,10,18]],"date-time":"2023-10-18T13:02:10Z","timestamp":1697634130000},"page":"1476-1498","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["An Identity for the Coefficients of Characteristic Polynomials of Hyperplane Arrangements"],"prefix":"10.1007","volume":"70","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8483-3373","authenticated-orcid":false,"given":"Zakhar","family":"Kabluchko","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,10,18]]},"reference":[{"key":"577_CR1","doi-asserted-by":"crossref","unstructured":"Amelunxen, D., Lotz, M.: Intrinsic volumes of polyhedral cones: a combinatorial perspective. Discrete Comput. 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