Bisecting envelopes of convex polygons. (English) Zbl 1486.52013
Summary: We introduce a new subject – bisecting deltoids – by giving status to two curves, the area-bisecting deltoid and the perimeter-bisecting deltoid of a convex polygon, of which the second is a new curve. These are the envelopes of all lines that bisect either the area or the perimeter of the polygon. The subject is located between the Euclidean and convex geometries in the same way as the elliptic and hyperbolic geometries are located between the Euclidean and differential geometries. We show that these curves have numerous geometric, differential, analytic, topological and combinatorial properties.
From the direction of the Euclidean, differential and analytic geometries, we describe the two curves both as geometric and analytic loci. Particular emphasis is given to deltoids of quadrilaterals, which are classified, and those of regular polygons. Our results here extend the known results of D. Ball [“Halving envelopes”, Math. Gaz. 64, No. 429, 166–173 (1980; doi:10.2307/3615118)], A. Berele and S. Catoiu [Math. Mag. 91, No. 2, 121–133 (2018; Zbl 1407.51007)] and J. A. Dunn and J. E. Pretty [Math. Gaz. 56, 105–108 (1972; Zbl 0234.50009)] on bisecting deltoids of triangles and those of N. Fechtor-Pradines [Involve 8, No. 2, 307–328 (2015; Zbl 1316.51016)] and R. Guàrdia and F. Hurtado [J. Geom. 83, No. 1–2, 32–45 (2005; Zbl 1089.52004)] on area-bisecting deltoids of convex polygons.
From the direction of convex geometry and combinatorial topology, we describe the area- or perimeter-bisecting partitions, or the incidence partitions – two partitions of the plane that are common to these curves. The area (perimeter) bisecting partition is the partition of the plane given by labeling each point by the number lines through it that bisect the area (perimeter) of the polygon. The same label is achieved by counting the number of tangents to the area (perimeter) bisecting deltoid that pass through the point. The bisecting partition is different from the geometric partition, which is common to any closed curve in the plane. Our description of the bisecting partition extends the known results of J. G. Ceder [Can. J. Math. 16, 1–11 (1964; Zbl 0117.39103)] and [Guàrdia and Hurtado, loc. cit.]. This is made possible by a new counting technique, the total sweep count.
From the direction of the Euclidean, differential and analytic geometries, we describe the two curves both as geometric and analytic loci. Particular emphasis is given to deltoids of quadrilaterals, which are classified, and those of regular polygons. Our results here extend the known results of D. Ball [“Halving envelopes”, Math. Gaz. 64, No. 429, 166–173 (1980; doi:10.2307/3615118)], A. Berele and S. Catoiu [Math. Mag. 91, No. 2, 121–133 (2018; Zbl 1407.51007)] and J. A. Dunn and J. E. Pretty [Math. Gaz. 56, 105–108 (1972; Zbl 0234.50009)] on bisecting deltoids of triangles and those of N. Fechtor-Pradines [Involve 8, No. 2, 307–328 (2015; Zbl 1316.51016)] and R. Guàrdia and F. Hurtado [J. Geom. 83, No. 1–2, 32–45 (2005; Zbl 1089.52004)] on area-bisecting deltoids of convex polygons.
From the direction of convex geometry and combinatorial topology, we describe the area- or perimeter-bisecting partitions, or the incidence partitions – two partitions of the plane that are common to these curves. The area (perimeter) bisecting partition is the partition of the plane given by labeling each point by the number lines through it that bisect the area (perimeter) of the polygon. The same label is achieved by counting the number of tangents to the area (perimeter) bisecting deltoid that pass through the point. The bisecting partition is different from the geometric partition, which is common to any closed curve in the plane. Our description of the bisecting partition extends the known results of J. G. Ceder [Can. J. Math. 16, 1–11 (1964; Zbl 0117.39103)] and [Guàrdia and Hurtado, loc. cit.]. This is made possible by a new counting technique, the total sweep count.
MSC:
| 52A30 | Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.) |
| 52A38 | Length, area, volume and convex sets (aspects of convex geometry) |
| 51N20 | Euclidean analytic geometry |
| 53A04 | Curves in Euclidean and related spaces |
Keywords:
area-bisecting deltoid; area-bisecting envelope; area-bisecting line; area-bisecting partition; deltoid; incidence partition; perimeter-bisecting deltoid; perimeter-bisecting line; perimeter-bisecting envelope; perimeter-bisecting partitionReferences:
| [1] | Arnold, V. I., Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, vol. 60 (1997), Springer-Verlag: Springer-Verlag New York |
| [2] | Aronov, B.; Goodman, J. E.; Pollack, R., A Helly-type theorem for higher-dimensional transversals, Comput. Geom., 21, 177-183 (2002) · Zbl 1009.52013 |
| [3] | Auerbach, H., Sur un problème de M. Ulam concernant l’équilibre des corps flottants, Stud. Math., 7, 121-142 (1938) · Zbl 0018.17504 |
| [4] | Avis, D., On the partitionability of point sets in space, Proc. First ACM Symposium on Computational Geometry, 116-120 (1985) |
| [5] | Ball, D., Halving envelopes, Math. Gaz., 64, 166-173 (1980) |
| [6] | Bárány, I.; Blagojević, P.; Szücs, A., Equipartitioning by a convex 3-fan, Adv. Math., 223, 579-593 (2010) · Zbl 1190.52001 |
| [7] | Bárány, I.; Hug, D.; Schneider, R., Affine diameters of convex bodies, Proc. Am. Math. Soc., 144, 797-812 (2016) · Zbl 1335.52006 |
| [8] | Bárány, I.; Hubard, A.; Jerónimo, J., Slicing convex sets and measures by a hyperplane, Discrete Comput. Geom., 39, 67-75 (2008) · Zbl 1142.52004 |
| [9] | Bárány, I.; Matoušek, J., Simultaneous partitions of measures by k-fans, Discrete Comput. Geom., 25, 317-334 (2001) · Zbl 0989.52009 |
| [10] | Bárány, I.; Matoušek, J., Equipartition of two measures by a 4-fan, Discrete Comput. Geom., 27, 293-301 (2002) · Zbl 1002.60006 |
| [11] | Bereg, S., Equipartitions of measures by 2-fans, Discrete Comput. Geom., 34, 87-96 (2005) · Zbl 1073.52003 |
| [12] | Berele, A.; Catoiu, S., Area and perimeter bisecting lines of a triangle, Coll. Math. J., 47, 19-28 (2016) · Zbl 1442.97014 |
| [13] | Berele, A.; Catoiu, S., The perimeter sixpartite center of a triangle, J. Geom., 108, 861-868 (2017) · Zbl 1384.51034 |
| [14] | Berele, A.; Catoiu, S., The centroid as a nontrivial area bisecting center of a triangle, Coll. Math. J., 49, 27-34 (2018) · Zbl 1407.97008 |
| [15] | Berele, A.; Catoiu, S., Bisecting the perimeter of a triangle, Math. Mag., 91, 121-133 (2018) · Zbl 1407.51007 |
| [16] | Berele, A.; Catoiu, S., Nonuniqueness of sixpartite points, Am. Math. Mon., 125, 638-642 (2018) · Zbl 1395.52002 |
| [17] | Berele, A.; Catoiu, S., The Fermat-Torricelli theorem in convex geometry, J. Geom., 111, 2, Article 22 pp. (2020), 21 pp. · Zbl 1437.52001 |
| [18] | Berele, A.; Catoiu, S., The classification of convex polygons with triangular area or perimeter bisecting deltoids, Beitr. Algebra Geom., 63, 95-114 (2022) · Zbl 1485.52002 |
| [19] | Berele, A.; Catoiu, S., Area computation for triangular area or perimeter bisecting deltoids, Aequ. Math. (2022), in press, 13 pp. |
| [20] | Berele, A.; Catoiu, S., Zindler points of triangles, Math. Mag. (2022), in press · Zbl 1505.51005 |
| [21] | Berele, A.; Catoiu, S., The pentagonal pizza conjecture, Am. Math. Mon., 129 (2022), in press, 9 pp. · Zbl 1487.52012 |
| [22] | A. Berele, S. Catoiu, You can’t cut two pancakes with compass and straightedge, preprint. · Zbl 07889442 |
| [23] | A. Berele, S. Catoiu, Congruent area quadrisectors in a triangle, preprint. · Zbl 07832072 |
| [24] | Berele, A.; Goldman, J., Geometry Theorems and Constructions (2001), Prentice Hall |
| [25] | Berger, M., Geometry I-II (2009), UTX, Springer-Verlag: UTX, Springer-Verlag Berlin, Translation of the original French edition: Géométrie (vols. 1-5), published by CEDIC, Fernand Nathan, Paris, 1977 |
| [26] | Besicovitch, A. S., On Kakeya’s problem and a similar one, Math. Z., 27, 312-320 (1928) · JFM 53.0713.01 |
| [27] | Besicovitch, A. S., Measure of asymmetry of convex curves, J. Lond. Math. Soc., 64, 237-240 (1948) · Zbl 0035.38402 |
| [28] | Besicovitch, A. S., The Kakeya problem, Am. Math. Mon., 70, 697-706 (1963) · Zbl 0117.39402 |
| [29] | Besicovitch, A. S.; Zamfirescu, T., On pencils of diameters in convex bodies, Rev. Roum. Math. Pures Appl., 11, 637-639 (1966) · Zbl 0146.44202 |
| [30] | Bespamyatnikh, S.; Kirkpatrick, D.; Snoeyink, J., Generalizing ham-sandwich cuts to equitable subdivisions, Discrete Comput. Geom., 24, 605-622 (2000) · Zbl 0966.68156 |
| [31] | Beyer, W. A.; Zardecki, A., The early history of the ham sandwich theorem, Am. Math. Mon., 111, 58-61 (2004) · Zbl 1047.55500 |
| [32] | Blagojević, P.; Ziegler, G., Convex equipartitions via equivariant obstruction theory, Isr. J. Math., 200, 49-77 (2014) · Zbl 1305.52005 |
| [33] | Blagojević, P.; Ziegler, G., Beyond the Borsuk-Ulam theorem: the topological Tverberg story, (A Journey Through Discrete Mathematics (2017), Springer: Springer Cham), 273-341 · Zbl 1470.52007 |
| [34] | Blaschke, W., Aufgaben 540 und 541, Arch. Math. Phys., 26, 65 (1917) |
| [35] | Blaschke, W., Kreis und Kugel (Circle and Sphere) (1956), W. de Gruyter: W. de Gruyter Berlin · Zbl 0070.17501 |
| [36] | Böhringer, K. F.; Donald, B. R.; Halperin, D., On the area bisectors of a polygon, Discrete Comput. Geom., 22, 269-285 (1999) · Zbl 0942.68130 |
| [37] | Born, M., Principle of Optics (1999), Cambridge University Press, Appendix I: The calculus of variations |
| [38] | Bottomley, H., Medians and area bisectors of a triangle (January 2002) |
| [39] | Breen, M., A Helly-type theorem for countable intersections of starshaped sets, Arch. Math. (Basel), 84, 282-288 (2005) · Zbl 1080.52006 |
| [40] | Breen, M., A Helly theorem for intersections of orthogonally starshaped sets, Arch. Math. (Basel), 80, 664-672 (2003) · Zbl 1034.52006 |
| [41] | Bruce, J. W.; Giblin, P. J., Curves and Singularities. A Geometrical Introduction to Singularity Theory (1984), Cambridge University Press · Zbl 0534.58008 |
| [42] | Buck, R. C.; Buck, Ellen F., Equipartition of convex sets, Math. Mag., 22, 195-198 (1949) |
| [43] | Burns, J. C., Construction of a line through a given point to divide a triangle into two parts with areas in a given ratio, Elem. Math., 41, 58-67 (1986) · Zbl 0589.51030 |
| [44] | Ceder, Jack J., Generalized sixpartite problems, Bol. Soc. Math. Mex. (2), 9, 28-32 (1964) · Zbl 0158.19802 |
| [45] | Ceder, Jack J., On outwardly simple line families, Can. J. Math., 16, 1-11 (1964) · Zbl 0117.39103 |
| [46] | Conway, J.; Ryba, A., The Pascal mysticum demystified, Math. Intell., 34, 4-8 (2012) · Zbl 1262.51003 |
| [47] | Conway, J.; Ryba, A., A characterization of equilateral triangles and some consequences, Math. Intell., 36, 1-2 (2014) · Zbl 1311.51011 |
| [48] | Courant, R.; Robbins, H., What Is Mathematics? (1941), Oxford University Press: Oxford University Press New York · Zbl 0060.12302 |
| [49] | Coxeter, H. S.M.; Greitzer, S. L., Geometry Revisited, New Mathematical Library, vol. 19 (1967), Random House, Inc.: Random House, Inc. New York · Zbl 0166.16402 |
| [50] | Danzer, L.; Grünbaum, B.; Klee, V., Helly’s theorem and its relatives, (Proc. Sympos. Pure Math., vol. VII (1963), Amer. Math. Soc.: Amer. Math. Soc. Providence, R.I.), 101-180 · Zbl 0132.17401 |
| [51] | Dunn, J. A.; Pretty, J. E., Halving a triangle, Math. Gaz., 56, 105-108 (1972) · Zbl 0234.50009 |
| [52] | Edelsbrunner, H.; Waupotitsch, R., Computing a ham sandwich cut in two dimensions, J. Symb. Comput., 2, 171-178 (1986) · Zbl 0623.68058 |
| [53] | Eggleston, H. G., Some properties of triangles as extremal convex curves, J. Lond. Math. Soc., 28, 32-36 (1953) · Zbl 0050.16602 |
| [54] | Eggleston, H. G., Problems in Euclidean Space: Aplications of Convexity, International Series of Monographs on Pure and Applied Mathematics, vol. V (1957), Pergamon Press · Zbl 0083.38102 |
| [55] | Eggleston, H. G., Convexity, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 47 (1958), Cambridge Univ. Press · Zbl 0086.15302 |
| [56] | Eisenhart, L. P., A Treatise on the Differential Geometry of Curves and Surfaces (1960), Dover: Dover New York · Zbl 0090.37803 |
| [57] | Evans, L. C., Partial Differential Equations, Graduate Studies in Mathematics, vol. 19 (1998), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0902.35002 |
| [58] | Fechtor-Pradines, N., Bisection envelopes, Involve, 8, 307-328 (2015) · Zbl 1316.51016 |
| [59] | Forsyth, A. R., Theory of Differential Equations (1959), Dover Publications: Dover Publications New York · Zbl 0088.05802 |
| [60] | Fusco, N.; Pratelli, A., On a conjecture of Auerbach, J. Eur. Math. Soc., 13, 1633-1676 (2011) · Zbl 1227.49047 |
| [61] | Goldberg, M., On area-bisectors of plane convex sets, Am. Math. Mon., 70, 529-531 (1963) · Zbl 0118.17303 |
| [62] | Goodey, P., Area and perimeter bisectors of planar convex sets, (Integral Geometry and Convexity (2006), World Sci.), 29-35 · Zbl 1124.52003 |
| [63] | Grünbaum, B., Partitions of mass-distributions and of convex bodies by hyperplanes, Pac. J. Math., 10, 1257-1261 (1960) · Zbl 0101.14603 |
| [64] | Grünbaum, B., Measures of symmetry for convex sets, (Proc. Sympos. Pure Math., vol. VII (1963), Amer. Math. Soc.: Amer. Math. Soc. Providence, R.I.), 233-270 · Zbl 0142.20503 |
| [65] | Grünbaum, B., Continuous families of curves, Can. J. Math., 18, 529-537 (1966) · Zbl 0142.20404 |
| [66] | Guàrdia, R.; Hurtado, F., On the equipartition of plane convex bodies and convex polygons, J. Geom., 83, 32-45 (2005) · Zbl 1089.52004 |
| [67] | Gunther, G.; Wilker, J. B., The bisectrix of a tetrahedron, Mathematika, 39, 95-103 (1992) · Zbl 0767.51007 |
| [68] | Guo, Q.; Toth, G., Dual mean Minkowski measures and the Grünbaum conjecture for affine diameters, Pac. J. Math., 292, 117-137 (2018) · Zbl 1376.52001 |
| [69] | Hammer, P. C., The centroid of a convex body, Proc. Am. Math. Soc., 2, 522-525 (1951) · Zbl 0043.16301 |
| [70] | Hammer, P. C., Convex bodies associated with a convex body, Proc. Am. Math. Soc., 2, 781-793 (1951) · Zbl 0043.16302 |
| [71] | Hammer, P. C., Diameters of convex bodies, Proc. Am. Math. Soc., 5, 304-306 (1954) · Zbl 0057.38603 |
| [72] | Hammer, P. C.; Sobczyk, A., Planar line families I, Proc. Am. Math. Soc., 4, 226-233 (1953) · Zbl 0051.13502 |
| [73] | Hammer, P. C.; Sobczyk, A., Planar line families II, Proc. Am. Math. Soc., 4, 341-349 (1953) · Zbl 0053.12301 |
| [74] | Helly, E., Über Mengen konvexer Körper mit gemeinschaftlichen Punkten, Jahresber. Dtsch. Math.-Ver., 32, 175-176 (1923) · JFM 49.0534.02 |
| [75] | Helly, E., Über Systeme von abgeschlossenen Mengen mit gemeinschaftlichen Punkten, Monatshefte Math. Phys., 37, 281-302 (1930) · JFM 56.0499.03 |
| [76] | Honsberger, R., Episodes in Nineteenth and Twentieth Century Euclidean Geometry, NML, vol. 37 (1995), Mathematical Association of America · Zbl 0829.01001 |
| [77] | Humphreys, J. E., Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, vol. 29 (1990), Cambridge University Press · Zbl 0725.20028 |
| [78] | John, F., Partial Differential Equations, Applied Mathematical Sciences, vol. 1 (1991), Springer-Verlag · Zbl 0742.35001 |
| [79] | Johnson, R. A., Advanced Euclidean Geometry (1960), Dover: Dover New York · Zbl 0090.37301 |
| [80] | Kaneko, A.; Kano, M., Balanced partitions of two sets of points in the plane, Comput. Geom., 13, 253-261 (1999) · Zbl 0948.68199 |
| [81] | Kazarinoff, N. D., Geometric Inequalities, New Mathematical Library, vol. 4 (1961), Mathematical Association of America |
| [82] | Klee, V., The critical set of a convex body, Am. J. Math., 75, 178-188 (1953) · Zbl 0050.16604 |
| [83] | Kryžanovskiĭ, D. A., Izoperimetry (Isoperimetry) (1938), Moscow-Leningrad |
| [84] | Lo, C-Y.; Steiger, W. L., An optimal time algorithm for ham-sandwish cuts in the plane, (Proceedings of the 2nd Canadian Conference on Computational Geometry (1990)), 5-9 |
| [85] | Lo, C-Y.; Matoušek, J.; Steiger, W. L., Algorithms for Ham-Sandwish cuts, Discrete Comput. Geom., 11, 433-452 (1994) · Zbl 0806.68061 |
| [86] | Lockwood, E. H., A Book of Curves (1967), Cambridge University Press: Cambridge University Press New York, Reprinted 2007 |
| [87] | Topics: 3-Points of a Triangle, Euler’s Line, Bisecting Lines, geometry.puzzles, the Math Forum@Drexel, October-December 2001. |
| [88] | Martini, H.; Montejano, L.; Oliveros, D., Bodies with Constant Width. An Introduction to Convex Geometry with Applications (2019), Birkhäuser/Springer: Birkhäuser/Springer Cham · Zbl 1468.52001 |
| [89] | Matoušek, J., Using the Borsuk-Ulam Theorem, Lectures on Topological Methods in Combinatorics and Geometry (2003), Springer-Verlag: Springer-Verlag Berlin · Zbl 1016.05001 |
| [90] | Nandakumar, R.; Rao, R. N. Ramana, Fair partitions of polygons: an elementary introduction, Proc. Indian Acad. Sci. Math. Sci., 122, 459-467 (2012) · Zbl 1260.52013 |
| [91] | Neumann, B. H., On some affine invariants of closed convex regions, J. Lond. Math. Soc., 14, 262-272 (1939) · Zbl 0026.35901 |
| [92] | Neumann, B. H., On an invariant of plane regions and mass distributions, J. Lond. Math. Soc., 20, 226-237 (1945) · Zbl 0063.05928 |
| [93] | Polya, G., Mathematics and Plausible Reasoning, Vol. I, II (1954), Princeton University Press, Reprinted 1990 · Zbl 0056.24101 |
| [94] | Roy, S.; Steiger, W., Some combinatorial and algorithmic aspects of the Borsuk-Ulam theorem, Graphs Comb., 23, 333-341 (2007) · Zbl 1123.52007 |
| [95] | Sakai, T., Balanced convex partitions of measures in \(R^2\), Graphs Comb., 18, 169-192 (2002) · Zbl 1002.52002 |
| [96] | Salmon, G., Treatise on Conic Sections (1960), Chelsea Publishing Co.: Chelsea Publishing Co. New York |
| [97] | Schneider, R., Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and Its Applications, vol. 151 (2014), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1287.52001 |
| [98] | Schulman, L. J., An equipartition of planar sets, Discrete Comput. Geom., 9, 257-266 (1993) · Zbl 0779.52008 |
| [99] | Sholander, M., Proof of a conjecture of R. C. Buck and E. F. Buck, Math. Mag., 24, 7-10 (1950) · Zbl 0041.51203 |
| [100] | Soltan, V., Affine diameters of convex bodies-a survey, Expo. Math., 23, 47-63 (2005) · Zbl 1076.52001 |
| [101] | Steinhaus, H., A note on the ham sandwich theorem, Math. Pol., 9, 26-28 (1938) |
| [102] | Steiger, W.; Zhao, J., Generalized ham-sandwich cuts for well-separated point sets, (Proceedings of the 20th Canadian Conference on Computational Geometry. Proceedings of the 20th Canadian Conference on Computational Geometry, Montreal (August 2008)) |
| [103] | Steiger, W.; Zhao, J., Generalized ham-sandwich cuts, Discrete Comput. Geom., 44, 535-545 (2010) · Zbl 1206.52008 |
| [104] | Steiger, W.; Szegedy, M.; Zhao, J., Six-way equipartitioning by three lines in the plane, (Proceedings of the 22nd Canadian Conference on Computational Geometry. Proceedings of the 22nd Canadian Conference on Computational Geometry, Winnipeg (August 2010)) |
| [105] | Stone, A. H.; Tukey, J. W., Generalized “sandwich” theorems, Duke Math. J., 9, 356-359 (1942) · Zbl 0061.38405 |
| [106] | Süss, W., Über eine Affininvariante von Eibereichen, Arch. Math. (Basel), 1, 127-128 (1948/49) · Zbl 0031.27803 |
| [107] | Toth, G., Asymmetry of convex sets with isolated extreme points, Proc. Am. Math. Soc., 137, 287-295 (2009) · Zbl 1165.52005 |
| [108] | Toth, G., On the structure of convex sets with symmetries, Geom. Dedic., 143, 69-80 (2009) · Zbl 1189.52007 |
| [109] | Toth, G., Measures of Symmetry for Convex Sets and Stability, Universitext (2015), Springer: Springer Cham · Zbl 1335.52002 |
| [110] | Weisstein, Eric W., “Deltoid” from MathWorld-a Wolfram Web Resource |
| [111] | Weisstein, Eric W., “Envelope” from MathWorld-a Wolfram Web Resource |
| [112] | Deltoid curve (mathematics) - Wikipedia, the free encyclopedia |
| [113] | Envelope (mathematics) - Wikipedia, the free encyclopedia |
| [114] | Workman, W. P., Memoranda Mathematica (2016), Clarendon Press: Clarendon Press Oxford: University of California Libraries, Reprinted by the |
| [115] | Yaglom, I. M., Geometric Transformations III, Anneli Lax New Mathematical Library, vol. 24 (1973), Mathematical Association of America: Mathematical Association of America Washington, DC, Third printing 2002 · Zbl 1177.51003 |
| [116] | Yaglom, I. M.; Boltyanskiĭ, V. G., Convex Figures (1960), Holt, Rinehart and Winston: Holt, Rinehart and Winston New York, Translated by Paul J. Kelly and Lewis F. Walton |
| [117] | Yates, R. C., A Handbook on Curves and Their Properties (1947), J. W. Edwards: J. W. Edwards Ann Arbor, Mich. |
| [118] | Zamfirescu, T., On planar continuous families of curves, Can. J. Math., 21, 513-530 (1969) · Zbl 0193.22401 |
| [119] | Zamfirescu, T., Les courbes fermée doubles sans points triples associées à une famille continue (French), Isr. J. Math., 7, 69-89 (1969) · Zbl 0188.27203 |
| [120] | Zamfirescu, T., Sur les familles continues de courbes I (French), Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. (8), 42, 771-774 (1967) · Zbl 0173.24501 |
| [121] | Zamfirescu, T., Sur les familles continues de courbes II (French), Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. (8), 43, 13-17 (1967) · Zbl 0173.24501 |
| [122] | Zamfirescu, T., Sur les familles continues de courbes III (French), Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. (8), 44, 639-642 (1968) · Zbl 0167.20402 |
| [123] | Zamfirescu, T., Sur les familles continues de courbes IV (French), Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. (8), 44, 753-758 (1968) · Zbl 0167.20403 |
| [124] | Zamfirescu, T., Sur les familles continues de courbes V (French), Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. (8), 53, 505-507 (1972) · Zbl 0282.54021 |
| [125] | Zamfirescu, T., On continuous families of curves VI, Geom. Dedic., 10, 205-217 (1981) · Zbl 0462.52002 |
| [126] | Zamfirescu, T.; Zucco, A., Continuous families of smooth curves and Grünbaum’s conjecture, Can. Math. Bull., 27, 345-350 (1984) · Zbl 0525.52012 |
| [127] | Zarankiewicz, K., Bisection of plane convex sets by lines (Polish), Wiad. Mat., 2, 228-234 (1959) · Zbl 0095.37002 |
| [128] | Zindler, K., Über konvexe Gebilde I, Monatshefte Math., 30, 87-102 (1920) · JFM 47.0681.06 |
| [129] | Zindler, K., Über konvexe Gebilde II, Monatshefte Math., 31, 25-56 (1921) · JFM 48.0833.05 |
| [130] | Zindler, K., Über konvexe Gebilde III, Monatshefte Math., 32, 107-138 (1922) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
