Pub Date : 2021-10-01DOI: 10.1515/advgeom-2021-0021
I. Herburt, S. Sakata
Abstract In this paper, we investigate an extremum problem for the power moment of a convex polygon contained in a disc. Our result is a generalization of a classical theorem: among all convex n-gons contained in a given disc, the regular n-gon inscribed in the circle (up to rotation) uniquely maximizes the area functional. It also implies that, among all convex n-gons contained in a given disc and containing the center in those interiors, the regular n-gon inscribed in the circle (up to rotation) uniquely maximizes the mean of the length of the chords passing through the center of the disc.
{"title":"An extremum problem for the power moment of a convex polygon contained in a disc","authors":"I. Herburt, S. Sakata","doi":"10.1515/advgeom-2021-0021","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0021","url":null,"abstract":"Abstract In this paper, we investigate an extremum problem for the power moment of a convex polygon contained in a disc. Our result is a generalization of a classical theorem: among all convex n-gons contained in a given disc, the regular n-gon inscribed in the circle (up to rotation) uniquely maximizes the area functional. It also implies that, among all convex n-gons contained in a given disc and containing the center in those interiors, the regular n-gon inscribed in the circle (up to rotation) uniquely maximizes the mean of the length of the chords passing through the center of the disc.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47885854","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-01DOI: 10.1515/advgeom-2020-0012
M. Giulietti, M. Kawakita, Stefano Lia, M. Montanucci
Abstract In 1895 Wiman introduced the Riemann surface 𝒲 of genus 6 over the complex field ℂ defined by the equation X6+Y6+ℨ6+(X2+Y2+ℨ2)(X4+Y4+ℨ4)−12X2Y2ℨ2 = 0, and showed that its full automorphism group is isomorphic to the symmetric group S5. We show that this holds also over every algebraically closed field 𝕂 of characteristic p ≥ 7. For p = 2, 3 the above polynomial is reducible over 𝕂, and for p = 5 the curve 𝒲 is rational and Aut(𝒲) ≅ PGL(2,𝕂). We also show that Wiman’s 𝔽192-maximal sextic 𝒲 is not Galois covered by the Hermitian curve H19 over the finite field 𝔽192.
{"title":"An 𝔽p2-maximal Wiman sextic and its automorphisms","authors":"M. Giulietti, M. Kawakita, Stefano Lia, M. Montanucci","doi":"10.1515/advgeom-2020-0012","DOIUrl":"https://doi.org/10.1515/advgeom-2020-0012","url":null,"abstract":"Abstract In 1895 Wiman introduced the Riemann surface 𝒲 of genus 6 over the complex field ℂ defined by the equation X6+Y6+ℨ6+(X2+Y2+ℨ2)(X4+Y4+ℨ4)−12X2Y2ℨ2 = 0, and showed that its full automorphism group is isomorphic to the symmetric group S5. We show that this holds also over every algebraically closed field 𝕂 of characteristic p ≥ 7. For p = 2, 3 the above polynomial is reducible over 𝕂, and for p = 5 the curve 𝒲 is rational and Aut(𝒲) ≅ PGL(2,𝕂). We also show that Wiman’s 𝔽192-maximal sextic 𝒲 is not Galois covered by the Hermitian curve H19 over the finite field 𝔽192.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41575118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-27DOI: 10.1515/advgeom-2022-0018
L. Kramer
Abstract We discuss various aspects of isometric group actions on proper metric spaces. As one application, we show that a proper and Weyl transitive action on a euclidean building is strongly transitive on the maximal atlas (the complete apartment system) of the building.
{"title":"Some remarks on proper actions, proper metric spaces, and buildings","authors":"L. Kramer","doi":"10.1515/advgeom-2022-0018","DOIUrl":"https://doi.org/10.1515/advgeom-2022-0018","url":null,"abstract":"Abstract We discuss various aspects of isometric group actions on proper metric spaces. As one application, we show that a proper and Weyl transitive action on a euclidean building is strongly transitive on the maximal atlas (the complete apartment system) of the building.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49458195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-03DOI: 10.1515/advgeom-2022-0006
E. Morales-Amaya, J. Jer'onimo-Castro, D. J. Verdusco Hernández
Abstract We prove the following result: Let K be a strictly convex body in the Euclidean space ℝn, n ≥ 3, and let L be a hypersurface which is the image of an embedding of the sphere 𝕊n–1, such that K is contained in the interior of L. Suppose that, for every x ∈ L, there exists y ∈ L such that the support cones of K with apexes at x and y differ by a central symmetry. Then K and L are centrally symmetric and concentric.
{"title":"A characterization of centrally symmetric convex bodies in terms of visual cones","authors":"E. Morales-Amaya, J. Jer'onimo-Castro, D. J. Verdusco Hernández","doi":"10.1515/advgeom-2022-0006","DOIUrl":"https://doi.org/10.1515/advgeom-2022-0006","url":null,"abstract":"Abstract We prove the following result: Let K be a strictly convex body in the Euclidean space ℝn, n ≥ 3, and let L be a hypersurface which is the image of an embedding of the sphere 𝕊n–1, such that K is contained in the interior of L. Suppose that, for every x ∈ L, there exists y ∈ L such that the support cones of K with apexes at x and y differ by a central symmetry. Then K and L are centrally symmetric and concentric.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49426726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-01DOI: 10.1515/advgeom-2021-0016
Liping Yuan, T. Zamfirescu, Yanxue Zhang
Abstract A cage is the 1-skeleton of a convex polytope in ℝ3. A cage is said to hold a set if the set cannot be continuously moved to a distant location, remaining congruent to itself and disjoint from the cage. In how many positions can (compact 2-dimensional) unit discs be held by a tetrahedral cage? We completely answer this question for all tetrahedra.
{"title":"Tetrahedral cages for unit discs","authors":"Liping Yuan, T. Zamfirescu, Yanxue Zhang","doi":"10.1515/advgeom-2021-0016","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0016","url":null,"abstract":"Abstract A cage is the 1-skeleton of a convex polytope in ℝ3. A cage is said to hold a set if the set cannot be continuously moved to a distant location, remaining congruent to itself and disjoint from the cage. In how many positions can (compact 2-dimensional) unit discs be held by a tetrahedral cage? We completely answer this question for all tetrahedra.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/advgeom-2021-0016","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41513026","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-01DOI: 10.1515/advgeom-2021-0012
R. Schwartz
Abstract We prove an integral formula for continuous paths of rectangles inscribed in a piecewise smooth loop. We use this integral formula to prove the inequality M(γ) ≥ Δ(γ)/2 – 1, where M(γ) denotes the total multiplicity of rectangle coincidences, i.e. pairs, triples, etc. of isometric rectangles inscribed in γ, and Δ(γ) denotes the number of stable diameters of γ, i.e. critical points of the distance function on γ.
{"title":"Inscribed rectangle coincidences","authors":"R. Schwartz","doi":"10.1515/advgeom-2021-0012","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0012","url":null,"abstract":"Abstract We prove an integral formula for continuous paths of rectangles inscribed in a piecewise smooth loop. We use this integral formula to prove the inequality M(γ) ≥ Δ(γ)/2 – 1, where M(γ) denotes the total multiplicity of rectangle coincidences, i.e. pairs, triples, etc. of isometric rectangles inscribed in γ, and Δ(γ) denotes the number of stable diameters of γ, i.e. critical points of the distance function on γ.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/advgeom-2021-0012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49023497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-01DOI: 10.1515/advgeom-2021-0018
M. de Boeck, G. Van de Voorde
Abstract A Kakeya set 𝓚 in an affine plane of order q is the point set covered by a set 𝓛 of q + 1 pairwise non-parallel lines. By Dover and Mellinger [6], Kakeya sets with size at least q2 – 3q + 9 contain a large knot, i.e. a point of 𝓚 lying on many lines of 𝓛. We improve on this result by showing that Kakeya set of size at least ≈ q2 – q q $begin{array}{} displaystyle sqrt{q} end{array}$ + 32 $begin{array}{} displaystyle frac{3}{2} end{array}$q contain a large knot, and we obtain a sharp result for planes containing a Baer subplane.
{"title":"A note on large Kakeya sets","authors":"M. de Boeck, G. Van de Voorde","doi":"10.1515/advgeom-2021-0018","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0018","url":null,"abstract":"Abstract A Kakeya set 𝓚 in an affine plane of order q is the point set covered by a set 𝓛 of q + 1 pairwise non-parallel lines. By Dover and Mellinger [6], Kakeya sets with size at least q2 – 3q + 9 contain a large knot, i.e. a point of 𝓚 lying on many lines of 𝓛. We improve on this result by showing that Kakeya set of size at least ≈ q2 – q q $begin{array}{} displaystyle sqrt{q} end{array}$ + 32 $begin{array}{} displaystyle frac{3}{2} end{array}$q contain a large knot, and we obtain a sharp result for planes containing a Baer subplane.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44795249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-28DOI: 10.1515/advgeom-2021-0013
D. Bartoli, M. Montanucci, F. Torres
Abstract Let 𝔽 be the finite field of order q2. It is sometimes attributed to Serre that any curve 𝔽-covered by the Hermitian curveHq+1:yq+1=xq+x ${{mathcal{H}}_{q+1}}:{{y}^{q+1}}={{x }^{q}}+x$is also 𝔽-maximal. For prime numbers q we show that every 𝔽-maximal curve x $mathcal{x}$of genus g ≥ 2 with | Aut(𝒳) | > 84(g − 1) is Galois-covered by Hq+1. ${{mathcal{H}}_{q+1}}.$The hypothesis on | Aut(𝒳) | is sharp, since there exists an 𝔽-maximal curve x $mathcal{x}$for q = 71 of genus g = 7 with | Aut(𝒳) | = 84(7 − 1) which is not Galois-covered by the Hermitian curve H72. ${{mathcal{H}}_{72}}.$
{"title":"𝔽p2-maximal curves with many automorphisms are Galois-covered by the Hermitian curve","authors":"D. Bartoli, M. Montanucci, F. Torres","doi":"10.1515/advgeom-2021-0013","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0013","url":null,"abstract":"Abstract Let 𝔽 be the finite field of order q2. It is sometimes attributed to Serre that any curve 𝔽-covered by the Hermitian curveHq+1:yq+1=xq+x ${{mathcal{H}}_{q+1}}:{{y}^{q+1}}={{x }^{q}}+x$is also 𝔽-maximal. For prime numbers q we show that every 𝔽-maximal curve x $mathcal{x}$of genus g ≥ 2 with | Aut(𝒳) | > 84(g − 1) is Galois-covered by Hq+1. ${{mathcal{H}}_{q+1}}.$The hypothesis on | Aut(𝒳) | is sharp, since there exists an 𝔽-maximal curve x $mathcal{x}$for q = 71 of genus g = 7 with | Aut(𝒳) | = 84(7 − 1) which is not Galois-covered by the Hermitian curve H72. ${{mathcal{H}}_{72}}.$","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43047106","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-24DOI: 10.1515/advgeom-2020-0024
V. Totik
Abstract We give a short, elementary and non-computational proof for the classical Beckman–Quarles theorem asserting that a map of a Euclidean space into itself that preserves distance 1 must be an isometry.
{"title":"The Beckman–Quarles theorem via the triangle inequality","authors":"V. Totik","doi":"10.1515/advgeom-2020-0024","DOIUrl":"https://doi.org/10.1515/advgeom-2020-0024","url":null,"abstract":"Abstract We give a short, elementary and non-computational proof for the classical Beckman–Quarles theorem asserting that a map of a Euclidean space into itself that preserves distance 1 must be an isometry.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43029542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-24DOI: 10.1515/advgeom-2021-0019
C. Fontanari
Abstract The moduli space M¯0,n(ℙ1,1) ${{bar{M}}_{0,n}}left( {{mathbb{P}}^{1}},1 right)$of n-pointed stable maps is a Mori dream space whenever the moduli space M¯0,n+3 of (n+3) ${{bar{M}}_{0,n+3}}; text{of} ;(n+3)$pointed rational curves is, and M¯0,n(ℙ1,1) ${{bar{M}}_{0,n}}left( {{mathbb{P}}^{1}},1 right)$is a log Fano variety for n ≤ 5.
{"title":"When is M0,n(ℙ1,1) a Mori dream space?","authors":"C. Fontanari","doi":"10.1515/advgeom-2021-0019","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0019","url":null,"abstract":"Abstract The moduli space M¯0,n(ℙ1,1) ${{bar{M}}_{0,n}}left( {{mathbb{P}}^{1}},1 right)$of n-pointed stable maps is a Mori dream space whenever the moduli space M¯0,n+3 of (n+3) ${{bar{M}}_{0,n+3}}; text{of} ;(n+3)$pointed rational curves is, and M¯0,n(ℙ1,1) ${{bar{M}}_{0,n}}left( {{mathbb{P}}^{1}},1 right)$is a log Fano variety for n ≤ 5.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/advgeom-2021-0019","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44645357","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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