A convex combinatorial property of compact sets in the plane and its roots in lattice theory

Document Type: Research Paper

Authors

1 Bolyai Institute, University of Szeged, Szeged, Aradi vértanúk tere 1, H6720 Hungary

2 Bolyai Institute, University of Szeged, Szeged, Aradi vértanúk tere 1, Hungary H6720

Abstract

K. Adaricheva and M. Bolat have recently proved that if $\,\mathcal U_0$ and $\,\mathcal U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $j\in \{0,1,2\}$ and $k\in\{0,1\}$ such that $\,\mathcal U_{1-k}$ is included in the convex hull of $\,\mathcal U_k\cup(\{A_0,A_1, A_2\}\setminus\{A_j\})$. One could say disks instead of circles.
Here we prove the existence of such a $j$ and $k$ for the more general case where $\,\mathcal U_0$ and $\,\mathcal  U_1$ are compact sets in the plane such that $\,\mathcal U_1$ is obtained from $\,\mathcal U_0$ by a positive homothety or by a translation.
Also, we give a short survey to show how lattice theoretical antecedents, including a series of papers on planar semimodular lattices by G. Grätzer and E. Knapp, lead to our result.

Keywords


[1] Adaricheva, K., Representing finite convex geometries by relatively convex sets, European  J. Combin. 37 (2014), 68-78.
[2] Adaricheva, K. and Bolat, M., Representation of convex geometries by circles on the plane, https://arxiv.org/pdf/1609.00092.
[3] Adaricheva, K. and Czédli, G., Note on the description of join-distributive lattices by permutations, Algebra Universalis 72(2) (2014), 155-162.
[4] Adaricheva, K.V., Gorbunov, V.A., and Tumanov, V.I., Join semidistributive lattices and convex geometries, Adv. Math. 173(1) (2003), 1-49.
[5] Adaricheva, K. and Nation, J.B., Convex geometries, in "Lattice Theory: Special Topics and Applications" 2, G. Grätzer and F. Wehrung, eds., Birkhäuser, 2015.
[6] Bogart, K.P., Freese, R., and Kung, J.P.S., "The Dilworth Theorems. Selected papers of Robert P. Dilworth", Birkhäuser, 1990.
[7] Bonnesen, T. and Fenchel, W., "Theory of convex bodies", Translated from the German and edited by L. Boron, C. Christenson, and B. Smith, BCS Associates,D Moscow, ID, 1987.
[8] Czédli, G., The matrix of a slim semimodular lattice, Order 29(1) (2012), 85-103.
[9] Czédli, G., Representing homomorphisms of distributive lattices as restrictions of congruences of rectangular lattices, Algebra Universalis 67(4) (2012), 313-345.
[10] Czédli, G., Coordinatization of join-distributive lattices, Algebra Universalis 71(4) (2014), 385-404.
[11] Czédli, G., Finite convex geometries of circles, Discrete Math. 330 (2014), 61-75.
[12] Czédli, G., Patch extensions and trajectory colorings of slim rectangular lattices, Algebra Universalis 72(2) (2014), 125-154.
[13] Czédli, G., A note on congruence lattices of slim semimodular lattices, Algebra Universalis 72(3) (2014), 225-230.
[14] Czédli, G., Characterizing circles by a convex combinatorial property, Acta Sci. Math. (Szeged) 83(3-4) (2017), 683-701.
[15] Czédli, G., An easy way to a theorem of Kira Adaricheva and Madina Bolat on convexity and circles, Acta Sci. Math. (Szeged) 83(3-4) (2017), 703-712.
[16] Czédli, G., Celebrating professor George A. Grätzer, Categories and General Algebraic Structures with Applications,D http://cgasa.sbu.ac.ir/data/cgasa/news/Gratzer.pdf.
[17] Czédli, G., An interview with George A. Grätzer, Categories and General Algebraic Structures with Applications,D http://cgasa.sbu.ac.ir/data/cgasa/news/czedli-gratzer_interview- 2018june1.pdf.
[18] Czédli, G., Circles and crossing planar compact convex sets, submitted to Acta Sci. Math. (Szeged), https://arxiv.org/pdf/1802.06457.
[19] Czédli, G. and Grätzer, G., Notes on planar semimodular lattices VII, Resections of planar semimodular lattices, Order 30(3) (2013), 847-858.
[20] Czédli, G. and Grätzer, G., Planar semimodular lattices: structure and diagrams, in "Lattice Theory: Special Topics and Applications" 1, Birkhäuser/Springer, 2014,D 91-130.
[21] Czédli, G., Grätzer, G., and Lakser, H., Congruence structure of planar semimodularD lattices: the General Swing Lemma, Algebra Universalis (2018, online), https://doi.org/10.1007/s00012-018-0483-2.
[22] Czédli, G. and Kincses, J., Representing convex geometries by almost-circles, Acta Sci. Math. (Szeged) 83(3-4) (2017), 393-414.
[23] Czédli, G. and Makay, G.: Swing lattice game and a short proof of the swing lemma for planar semimodular lattices, Acta Sci. Math. (Szeged) 83(1-2) (2017), 13-29.
[24] Czédli, G., Ozsvárt, L., and Udvari, B., How many ways can two composition seriesD intersect?, Discrete Math. 312(24) (2012), 3523-3536.
[25] Czédli, G. and Schmidt, E.T., The Jordan-Hölder theorem with uniqueness for groups and semimodular lattices, Algebra Universalis 66(1-2) (2011), 69-79.
[26] Czédli, G. and Schmidt, E.T., Slim semimodular lattices I, A visual approach, OrderD 29(3) (2012), 481-497.
[27] Czédli, G. and Schmidt, E.T., Slim semimodular lattices II, A description by patchworkD systems, Order 30(2) (2013), 689-721.
[28] Czédli, G. and Stachó, L.L., A note and a short survey on supporting lines of compact convex sets in the plane, Acta Univ. M. Belii Ser. Math. 24 (2016), 3-14.
[29] Dilworth, R.P., Lattices with unique irreducible decompositions, Ann. of Math. 41(4) (1940), 771-777.
[30] Edelman, P.H., Meet-distributive lattices and the anti-exchange closure, Algebra Universalis 10(3) (1980), 290-299.
[31] Edelman, P.H. and Jamison, R.E., The theory of convex geometries, Geom. Dedicata 19(3) (1985), 247-271.
[32] Erdos, P. and Straus, E.G., Über eine geometrische Frage von Fejes-Tóth, Elem. Math. 23 (1968), 11-14.
[33] Fejes-Tóth, L., Eine Kennzeichnung des Kreises, Elem. Math. 22 (1967), 25-27.
[34] Freese, R., Ježek, J., and Nation, J.B., "Free lattices", Mathematical Surveys and Monographs 42, American Mathematical Society, 1995.
[35] Funayama, N., Nakayama, T., On the distributivity of a lattice of lattice-congruences, Proc. Imp. Acad. Tokyo 18 (1942), 553-554.
[36] Grätzer, G., "The Congruences of a Finite Lattice. A Proof-by-picture Approach", Birkhäuser, 2006.
[37] Grätzer, G., Planar semimodular lattices: congruences, in "Lattice Theory: Special Topics and Applications" 1, Birkhäuser/Springer, 2014, 131-165.
[38] Grätzer, G., Congruences in slim, planar, semimodular lattices: the swing lemma, Acta Sci. Math. (Szeged) 81(3-4) (2015), 381-397.
[39] Grätzer, G., On a result of Gábor Czédli concerning congruence lattices of planar semimodular lattices, Acta Sci. Math. (Szeged) 81(1-2) (2015), 25-32.
[40] Grätzer, G., "The Congruences of a Finite Lattice. A Proof-by-picture Approach", Birkhäuser/Springer, 2016.
[41] Grätzer, G. and Knapp, E., Notes on planar semimodular lattices I, Construction, Acta Sci. Math. (Szeged) 73(3-4) (2007), 445-462.
[42] Grätzer, G. and Knapp, E., Notes on planar semimodular lattices II, Congruences, Acta Sci. Math. (Szeged) 74(1-2) (2008), 37-47.
[43] Grätzer, G. and Knapp, E., A note on planar semimodular lattices, Algebra Universalis 58(4) (2008), 497-499.
[44] Grätzer, G. and Knapp, E., Notes on planar semimodular lattices III, Congruences of rectangular lattices, Acta Sci. Math. (Szeged) 75(1-2) (2009), 29-48.
[45] Grätzer, G. and Knapp, E., Notes on planar semimodular lattices IV, The size of a minimal congruence lattice representation with rectangular lattices, Acta Sci. Math. (Szeged) 76(1-2) (2010), 3-26.
[46] Grätzer, G. and Nation, J.B., A new look at the Jordan-Hölder theorem for semimodular lattices, Algebra Universalis 64(3-4) (2010), 309-311.
[47] Grätzer, G. and Schmidt, E.T., On congruence lattices of lattices, Acta Math. Acad. Sci. Hungar. 13(1-2) (1962), 179-185.
[48] Grätzer, G. and Schmidt, E.T., An extension theorem for planar semimodular lattices, Period. Math. Hungar. 69(1) (2014), 32-40.
[49] Grätzer, G. and Schmidt, E.T., A short proof of the congruence representation theorem of rectangular lattices, Algebra Universalis 71(1) (2014), 65-68.
[50] Hüsseinov, F., A note on the closedness of the convex hull and its applications, Journal of Convex Analysis 6(2) (1999), 387-393.
[51] Jónsson, B. and Nation, J.B., A report on sublattices of a free lattice, in "Contributions to Universal Algebra", Colloq. Math. Soc. János Bolyai 17, North-Holland, 1977, 223-257.
[52] Kashiwabara, K., Nakamura, M., and Okamoto, Y., The affine representation theorem for abstract convex geometries, Comput. Geom. 30(2) (2005), 129-144.
[53] Kincses, J.: On the representation of finite convex geometries with convex sets, Acta Sci. Math. (Szeged) 83(1-2) (2017), 301-312.
[54] Latecki, L., Rosenfeld., A., and Silverman, R., Generalized convexity: CP3 and boundaries of convex sets, Pattern Recognition 28 (1995), 1191-1199.
[55] Monjardet, B., A use for frequently rediscovering a concept, Order 1(4) (1985), 415-D 417.
[56] Richter, M. and Rogers, L.G., Embedding convex geometries and a bound on convex dimension, Discrete Math. 340(5) (2017), 1059-1063.
[57] Schneider, R., "Convex Bodies: The Brunn-Minkowski Theory", Encyclopedia of mathematics and its applications 44, Cambridge University Press, 1993.
[58] Toponogov, V.A., "Differential Geometry of Curves and Surfaces, A Concise Guide", Birkhäuser, 2006.
[59] Wehrung, F., A solution to Dilworth’s congruence lattice problem, Adv. Math. 216(2) (2007), 610-625.
[60] Yaglom, I.M. and Boltyanskiˇı, V.G., "Convex Figures", English translation, Holt, Rinehart and Winston Inc., 1961.