Support and separation properties of convex sets in finite dimension

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Valeriu Soltan

Abstract

This is a survey on support and separation properties of convex sets in the n-dimensional Euclidean space. It contains a detailed account of existing results, given either chronologically or in related groups, and exhibits them in a uniform way, including terminology and notation. We first discuss classical Minkowski’s theorems on support and separation of convex bodies, and next describe various generalizations of these results to the case of arbitrary convex sets, which concern bounding and asymptotic hyperplanes, and various types of separation by hyperplanes, slabs, and complementary convex sets.

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How to Cite
Soltan, V. (2021). Support and separation properties of convex sets in finite dimension. Extracta Mathematicae, 36(2), 241-278. https://doi.org/10.17398/2605-5686.36.2.241
Section
Convex Geometry

References

[1] A. Auslender, M. Teboulle, “Asymptotic Cones and Functions in Optimization and Variational Inequalities”, Springer, New York, 2003.
[2] J. Bair, J. Gwinner, Sur la séparation vraie de cônes convexes, Arkiv Mat. 16 (1978), 207 – 212.
[3] J. Bair, F. Jongmans, La séparation vraie dans un espace vectoriel, Bull. Soc. Roy. Sci. Liège 41 (1972), 163 – 170.
[4] F. Bernstein, Über das Gaußsche Fehlergesetz, Math. Ann. 64 (1907), 417 – 448.
[5] F. Bernstein, Konvexe Kurven mit überall dichter Menge von Ecken, Arch.
Math. Phys. 12 (1907), 285 – 286.
[6] L. Bieberbach, “Differentialgeometrie”, Teubner, Leipzig, 1932.
[7] W. Blaschke, “Kreis und Kugel”, Viet & Co., Leipzig, 1916.
[8] T. Bonnesen, W. Fenchel, “Theorie der konvexen Körper”, Springer, Berlin, 1934. English translation: “Theory of Convex Bodies”, BCS Associates, Moscow, ID, 1987.
[9] T. Botts, Convex sets, Amer. Math. Monthly 49 (1942), 527 – 535.
[10] H. Brunn, Zur Theorie der Eigebiete, Arch. Math. Phys. 17 (1911), 289 – 300.
[11] H. Brunn, Fundamentalsatz von den Stützen eines Eigebietes, Math. Ann. 100 (1928), 634 – 637.
[12] H. Brunn, Vom Normalenkegel der Zwischenebenen zweier getrennter Eikörper, S.-B. Bayer. Akad. der Wiss. Math.-Natur. Klasse, München (1930), 165 – 182.
[13] Yu.D. Burago, V.A. Zalgaller, Sufficient conditions for convexity, J. Soviet Math. 16 (1978), 395 – 434.
[14] D.G. Caraballo, Convexity, local simplicity, and reduced boundaries of sets, J. Convex Anal. 18 (2011), 823 – 832.
[15] D.G. Caraballo, Reduced boundaries and convexity, Proc. Amer. Math. Soc. 141 (2013), 1775 – 1782.
[16] C. Carathéodory, Über den Variabilitätsbereich der Koefficienten von Potenzreihen, die gegebene Werte nicht annehmen, Math. Ann. 64 (1907), 95 – 115.
[17] C. Carathéodory, Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen, Rend. Circ. Mat. Palermo 32 (1911), 193 – 217.
[18] S.N. Černikov, “Linear Inequalities” (Russian), Nauka, Moscow, 1968.
[19] G.B. Dantzig, “Linear Programming and Extensions”, Princeton University Press, Princeton, NJ, 1963.
[20] A. Dax, The distance between two convex sets, Linear Algebra Appl. 416 (2006), 184 – 213.
[21] M. De Wilde, Some properties of the exposed points of finite dimensional convex sets, J. Math. Anal. Appl. 99 (1984), 257 – 264.
[22] J.C. Dupin, G. Coquet, Caractérisations et propriétés des couples de convexes complémentaires, C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A273 – A276.
[23] J.C. Dupin, G. Coquet, Caractérisations et propriétés des couples de convexes complémentaires, C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A383 – A386.
[24] R. Durier, The Fermat-Weber problem and inner product spaces, J. Approx. Theory 78 (1994), 161 – 173.
[25] M. Eidelheit, Zur Theorie der konvexen Mengen in linearen normierten Räumen, Stud. Math. 6 (1936), 104 – 111.
[26] J.W. Ellis, A general set-separation theorem, Duke Math. J. 19 (1952), 417 – 421.
[27] M.D. Fajardo, M.A. Goberna, M.M.L. Rodrı́guez, J. Vicente-Pérez, “Even Convexity and Optimization”, Springer, 2020.
[28] J. Favard, Sur les corps convexes, J. Math. Pures Appl. 12 (1933), 219 – 282.
[29] W. Fenchel, A remark on convex sets and polarity, Comm. Sém. Math. Univ. Lund. Tome Suppl. (1952), 82 – 89.
[30] W. Fenchel, “Convex Cones, Sets, and Functions. Mimeographed Lecture Notes. Spring Term 1951”, Princeton University, Princeton, NJ, 1953.
[31] M. Fujiwara, Über die Anzahl der Kantenlinien einer geschlossenen konvexen Fläche, Tôhoku Math. J. 10 (1916), 164 – 166.
[32] Z.R. Gabidullina, A linear separability criterion for sets of Euclidean space, J. Optim. Theory Appl. 158 (2013), 145 – 171.
[33] D. Gale, V. Klee, Continuous convex sets, Math. Scand. 7 (1959), 379 – 391.
[34] H. Gericke, Über ein Konvexitätskriterium, Math. Z. 43 (1937), 110 – 112.
[35] M.A. Goberna, E. González, J.E. Martı́nez-Legaz, M.I. Todorov, Motzkin decomposition of closed convex sets, J. Math. Anal. Appl. 364 (2010), 209 – 221.
[36] V.V. Gorokhovik, E.A. Semenkova, Step-linear functions in finite-dimensional vector spaces. Definition, properties and their relation to half-spaces, (Russian) Dokl. Akad. Nauk Belarusi 41 (1997), 10 – 14.
[37] O. Güler, “Foundations of Optimization”, Springer, New York, 2010.
[38] B.P. Haalmeijer, On convex regions, Nieuw Arch. Wisk. 12 (1917), 152 – 160.
[39] P.C. Hammer, Maximal convex sets, Duke Math. J. 22 (1955), 103 – 106.
[40] P.C. Hammer, Semispaces and the topology of convexity, in “Convexity” (edited by V-L. Klee), Amer. Math. Soc., Providence, RI, 1963, 305 – 316.
[41] A.N. Iusem, J.E. Martı́nez-Legaz, M.I. Todorov, Motzkin predecomposable sets, J. Global Optim. 60 (2014), 635 – 647.
[42] R.E. Jamison, Some intersection and generation properties of convex sets, Compos. Math. 35 (1977), 147 – 161.
[43] R.E. Jamison, The space of maximal convex sets, Fund. Math. 111 (1981), 45 – 59.
[44] J.L. Jensen, Sur les fonctions convexes et les inégalités entre les inégalités entre les valeurs moyeunes, Acta Math. 30 (1906), 175 – 193.
[45] S. Kakeya, On some properties of convex curves and surfaces, Tôhoku Math. J. 8 (1915), 218 – 221.
[46] V.L. Klee, Separation properties of convex cones, Proc. Amer. Math. Soc. 6 (1955), 313 – 318.
[47] V.L. Klee, Strict separation of convex sets, Proc. Amer. Math. Soc. 7 (1956), 735 – 737.
[48] V.L. Klee, The structure of semispaces, Math. Scand. 4 (1956), 54 – 64.
[49] V.L. Klee, Asymptotes and projections of convex sets, Math. Scand. 8 (1960), 356 – 362.
[50] V.L. Klee, Maximal separation theorems for convex sets, Trans. Amer. Math. Soc. 134 (1968), 133 – 147.
[51] V.L. Klee, Separation and support properties of convex sets–a survey, In: A.V. Balakrishnan (ed), “Control Theory and the Calculus of Variations”, Academic Press, New York, 1969, pp. 235 – 303.
[52] V.L. Klee, Sharper approximation of extreme points by far points, Arch. Math. (Basel) 60 (1993), 383 – 388.
[53] G. Köthe, “Topologische Lineare Räume. I”, Berlin, Springer, 1960.
[54] M. Lassak, Convex half-spaces, Fund. Math. 120 (1984), 7 – 13.
[55] M. Lassak, A. Prószynski, Translate-inclusive sets, orderings and convex half-spaces, Bull. Polish Acad. Sci. Math. 34 (1986), 195 – 201.
[56] M. Lassak, A. Prószynski, Algebraic and geometric approach to the classification of semispaces, Math. Scand. 61 (1987), 204 – 212.
[57] J. Lawrence, V. Soltan, On unions and intersections of nested families of cones, Beitr. Algebra Geom. 57 (2016), 655 – 665.
[58] K. Leichtweiß, “Konvexe Mengen”, Springer, Berlin, 1980.
[59] J.E. Martı́nez-Legaz, Exact quasiconvex conjugation, Z. Oper. Res. Ser. A-B 27 (1983), A257 – A266.
[60] J.E. Martı́nez-Legaz, I. Singer, Lexicographical separation in Rn , Linear Algebra Appl. 90 (1987), 147 – 163.
[61] J.E. Martı́nez-Legaz, I. Singer, The structure of hemispaces in Rn, Linear Algebra Appl. 110 (1988), 117 – 179.
[62] S. Mazur, Über konvexe Mengen in linearen normierten Räumen, Studia Math. 4 (1933), 70 – 84.
[63] H. Minkowski, “Geometrie der Zahlen. I”, Teubner, Leipzig, 1896; II. Teubner, Leipzig, 1910.
[64] H. Minkowski, “Gesammelte Abhandlungen. Bd 2”, Teubner, Leipzig, 1911.
[65] C.E. Moore, Concrete semispaces and lexicographic order, Duke Math. J. 40 (1973), 53 – 61.
[66] Th. Motzkin, “Linear Inequalities. Mimeographed Lecture Notes”, University of California, Los Angeles, CA, 1951.
[67] G. Nöbeling, Über die Konvexität von Raumstücken, Sitzungsber. Bayer. Akad. Wiss. Math.-Naturwiss. Kl. (1937), 63 – 67.
[68] Z. Páles, Separation theorems for convex sets and convex functions with invariance properties, Lecture Notes in Econom. and Math. Systems 502 (2001), 279 – 293.
[69] M.J. Panik, “Fundamentals of Convex Analysis”, Kluwer, Dordrecht, 1993.
[70] K. Reinhardt, Über einen Satz von Herrn H. Tietze, Jahresber. Deutsch. Math.-Ver. 38 (1929), 191 – 192.
[71] K. Reidemeister, Über die singulären Randpunkte eines konvexen Körpers, Math. Ann. 83 (1921), 116 – 118.
[72] R.T. Rockafellar, “Convex Analysis”, Princeton University Press, Princeton, NJ, 1970.
[73] I. Singer, Generalized convexity, functional hulls and applications to conjugate duality in optimization, Lecture Notes in Econom. and Math. Systems 226 (1984), 49 – 79.
[74] V. Soltan, Polarity and separation of cones, Linear Algebra Appl. 538 (2018), 212 – 224.
[75] V. Soltan, Asymptotic planes and closedness conditions, J. Convex Anal. 25 (2018), 1183 – 1196
[76] V. Soltan, “Lectures on Convex Sets. Second Edition”, World Scientific, Hackensack, NJ, 2020.
[77] V. Soltan, On M-decomposable sets, J. Math. Anal. Appl. 485 (2020), Paper No. 123816, 15 pp.
[78] V. Soltan, Asymmetric separation of convex sets, Bul. Acad. Ştiinţe Repub. Mold. Mat. 485 (2) (2020), 88 – 101.
[79] V. Soltan, On M-predecomposable sets, Beitr. Algebra Geom. 62 (2021), 205 – 218.
[80] V. Soltan, Penumbras and separation of convex set, Results Math. 76 (2021), Paper No. 25, 21 pp.
[81] V. Soltan, Cone asymptotes of convex sets, Extracta Math. 36 (2021), 81 – 98.
[82] V. Soltan, Separating hyperplanes of convex sets, J. Convex Anal. 28 (2021), no. 4, 20 pp.
[83] E. Steinitz, Bedingt konvergente Reihen und konvexe Systeme. I–IV, J. Reine Angew. Math. 143 (1913), 128 – 175.
[84] E. Steinitz, Bedingt konvergente Reihen und konvexe Systeme. VI–VII, J. Reine Angew. Math. 146 (1916), 1 – 52.
[85] M.H. Stone, “Convexity. Mimeographed Lectures. Fall Quarter 1946”, University of Chicago, Chicago, IL, 1946.
[86] S. Straszewicz, “Beiträge zur Theorie der Konvexen Punktmengen”, Inaugural Dissertation, Meier, Zürich, 1914.
[87] S. Straszewicz, Über exponierte Punkte abgeschlossener Punktmengen, Fund. Math. 24 (1935), 139 – 143.
[88] W. Süß, Eine Kennzeichnung von Eibereichen, Tôhoku Math. J. 32 (1930), 362 – 364.
[89] H. Tietze, Über konvexe Figuren, J. Reine Angew. Math. 158 (1927), 168 – 172.
[90] H. Tietze, Eine charakteristische Eigenschaft der abgeschlossenen konvexen Punktmengen, Math. Ann. 99 (1928), 394 – 398.
[91] J.W. Tukey, Some notes on the separation of convex sets, Portugal. Math. 3 (1942), 95 – 102.