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We show that there is a threshold value $$\\lambda _2 = \\sqrt{\\sqrt{7}\/2 - 1\/4} \\approx 1.035797\\ldots $$<\/jats:tex-math>\n \n \n \u03bb<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n =<\/mml:mo>\n \n \n \n 7<\/mml:mn>\n <\/mml:msqrt>\n \/<\/mml:mo>\n 2<\/mml:mn>\n -<\/mml:mo>\n 1<\/mml:mn>\n \/<\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msqrt>\n \u2248<\/mml:mo>\n 1.035797<\/mml:mn>\n \u2026<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, such that for $$\\lambda <\\lambda _2$$<\/jats:tex-math>\n \n \u03bb<\/mml:mi>\n <<\/mml:mo>\n \n \u03bb<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> the critical covering area $$A^*(\\lambda )$$<\/jats:tex-math>\n \n \n A<\/mml:mi>\n \u2217<\/mml:mo>\n <\/mml:msup>\n \n (<\/mml:mo>\n \u03bb<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is $$A^*(\\lambda )=3\\pi \\left( \\frac{\\lambda ^2}{16} +\\frac{5}{32} + \\frac{9}{256\\lambda ^2}\\right) $$<\/jats:tex-math>\n \n \n A<\/mml:mi>\n \u2217<\/mml:mo>\n <\/mml:msup>\n \n (<\/mml:mo>\n \u03bb<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n 3<\/mml:mn>\n \u03c0<\/mml:mi>\n \n \n \n \u03bb<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n 16<\/mml:mn>\n <\/mml:mfrac>\n +<\/mml:mo>\n \n 5<\/mml:mn>\n 32<\/mml:mn>\n <\/mml:mfrac>\n +<\/mml:mo>\n \n 9<\/mml:mn>\n \n 256<\/mml:mn>\n \n \u03bb<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:mfrac>\n <\/mml:mfenced>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and for $$\\lambda \\ge \\lambda _2$$<\/jats:tex-math>\n \n \u03bb<\/mml:mi>\n \u2265<\/mml:mo>\n \n \u03bb<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the critical area is $$A^*(\\lambda )=\\pi (\\lambda ^2+2)\/4$$<\/jats:tex-math>\n \n \n A<\/mml:mi>\n \u2217<\/mml:mo>\n <\/mml:msup>\n \n (<\/mml:mo>\n \u03bb<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \u03c0<\/mml:mi>\n \n (<\/mml:mo>\n \n \u03bb<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n +<\/mml:mo>\n 2<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n \/<\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>; these values are tight. For the special case $$\\lambda =1$$<\/jats:tex-math>\n \n \u03bb<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, i.e., for covering a unit square, the critical covering area is $$\\frac{195\\pi }{256}\\approx 2.39301\\ldots $$<\/jats:tex-math>\n \n \n \n 195<\/mml:mn>\n \u03c0<\/mml:mi>\n <\/mml:mrow>\n 256<\/mml:mn>\n <\/mml:mfrac>\n \u2248<\/mml:mo>\n 2.39301<\/mml:mn>\n \u2026<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The proof uses a careful combination of manual and automatic analysis, demonstrating the power of the employed interval arithmetic technique.<\/jats:p>","DOI":"10.1007\/s00454-023-00582-1","type":"journal-article","created":{"date-parts":[[2023,10,6]],"date-time":"2023-10-06T14:01:34Z","timestamp":1696600894000},"page":"1232-1283","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Worst-Case Optimal Covering of Rectangles by Disks"],"prefix":"10.1007","volume":"72","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9062-4241","authenticated-orcid":false,"given":"S\u00e1ndor P.","family":"Fekete","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Utkarsh","family":"Gupta","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Phillip","family":"Keldenich","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sahil","family":"Shah","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christian","family":"Scheffer","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,10,6]]},"reference":[{"issue":"06","key":"582_CR1","doi-asserted-by":"publisher","first-page":"685","DOI":"10.1142\/S021819591100386X","volume":"21","author":"AK Abu-Affash","year":"2011","unstructured":"Abu-Affash, A.K., Carmi, P., Katz, M.J., Morgenstern, G.: Multi cover of a polygon minimizing the sum of areas. 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