Some generalizations on affine invariant points

Jonard-Pérez, Natalia

doi:10.1090/proc/15229

Some generalizations on affine invariant points

Author: Natalia Jonard-Pérez
Journal: Proc. Amer. Math. Soc. 148 (2020), 5299-5311
MSC (2010): Primary 52A20, 54B20, 54H15, 57S20
DOI: https://doi.org/10.1090/proc/15229
Published electronically: September 11, 2020
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this note we prove a more general (and topological) version of Grünbaum's conjecture about affine invariant points. As an application of our result we show that if we consider the action of the group of similarities, Grünbaum's conjecture remains valid in other families of convex sets (not necessarily convex bodies).

[1] S. A. Antonjan, Retracts in categories of 𝐺-spaces, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 15 (1980), no. 5, 365–378, 417 (Russian, with English and Armenian summaries). MR 604847
[2] S. A. Antonyan, Equivariant generalization of Dugundji’s theorem, Mat. Zametki 38 (1985), no. 4, 608–616, 636 (Russian). MR 819426
[3] Sergey Antonian, An equivariant theory of retracts, Aspects of topology, London Math. Soc. Lecture Note Ser., vol. 93, Cambridge Univ. Press, Cambridge, 1985, pp. 251–269. MR 787832
[4] Sergey A. Antonyan, Extending equivariant maps into spaces with convex structure, Topology Appl. 153 (2005), no. 2-3, 261–275. MR 2175350, https://doi.org/10.1016/j.topol.2003.09.015
[5] Sergey A. Antonyan and Natalia Jonard-Pérez, Equivariant selections of convex-valued maps, Topology Proc. 40 (2012), 227–238. MR 2854082
[6] Sergey A. Antonyan and Natalia Jonard-Pérez, Affine group acting on hyperspaces of compact convex subsets of ℝⁿ, Fund. Math. 223 (2013), no. 2, 99–136. MR 3145542, https://doi.org/10.4064/fm223-2-1
[7] Sergey A. Antonyan, Natalia Jonard-Pérez, and Saúl Juárez-Ordóñez, Hyperspaces of convex bodies of constant width. part B, Topology Appl. 196 (2015), no. part B, 347–361. MR 3430982, https://doi.org/10.1016/j.topol.2014.05.027
[8] Glen E. Bredon, Introduction to compact transformation groups, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 46. MR 0413144
[9] D. van Dantzig and B. L. van der Waerden, Über metrisch homogene räume, Abh. Math. Sem. Univ. Hamburg 6 (1928), no. 1, 367–376 (German). MR 3069509, https://doi.org/10.1007/BF02940622
[10] Bernardo González Merino and Natalia Jonard-Pérez, A pseudometric invariant under similarities in the hyperspace of non-degenerated compact convex sets of ℝⁿ, Topology Appl. 194 (2015), 125–143. MR 3404607, https://doi.org/10.1016/j.topol.2015.08.002
[11] Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR 1039321
[12] Branko Grünbaum, Measures of symmetry for convex sets, Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, pp. 233–270. MR 0156259
[13] Charles J. Himmelberg, Some theorems on equiconnected and locally equiconnected spaces, Trans. Amer. Math. Soc. 115 (1965), 43–53. MR 195038, https://doi.org/10.1090/S0002-9947-1965-0195038-X
[14] Ivan Iurchenko, Affine invariant points and new constructions, J. Math. Anal. Appl. 445 (2017), no. 2, 1410–1416. MR 3545250, https://doi.org/10.1016/j.jmaa.2016.08.011
[15] Natalia Jonard-Pérez, A short proof of Grünbaum’s conjecture about affine invariant points, Topology Appl. 204 (2016), 240–245. MR 3482720, https://doi.org/10.1016/j.topol.2016.03.013
[16] P. A. Kučment, On the question of the affine-invariant points of convex bodies, Optimizacija 8(25) (1972), 48–51, 127 (Russian). MR 0350621
[17] P. Kuchment,
On a problem concerning affine-invariant points of convex bodies, (English translation of [16]),
1602.04377, 2016.
[18] Horst Martini, Luis Montejano, and Déborah Oliveros, Bodies of constant width, Birkhäuser/Springer, Cham, 2019. An introduction to convex geometry with applications. MR 3930585
[19] Mathieu Meyer, Carsten Schütt, and Elisabeth M. Werner, Affine invariant points, Israel J. Math. 208 (2015), no. 1, 163–192. MR 3416917, https://doi.org/10.1007/s11856-015-1196-2
[20] Olaf Mordhorst, New results on affine invariant points, Israel J. Math. 219 (2017), no. 2, 529–548. MR 3649598, https://doi.org/10.1007/s11856-017-1489-8
[21] Maria Moszyńska, Selected topics in convex geometry, Birkhäuser Boston, Inc., Boston, MA, 2006. Translated and revised from the 2001 Polish original. MR 2169492
[22] R. Palais,
The classification of -spaces,
Memoirs of the American Mathematical Society, vol 36, American Mathematical Society, Providence, RI, 1960.
[23] Richard S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2) 73 (1961), 295–323. MR 126506, https://doi.org/10.2307/1970335
[24] Katsuro Sakai, Geometric aspects of general topology, Springer Monographs in Mathematics, Springer, Tokyo, 2013. MR 3099433
[25] Roger Webster, Convexity, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994. MR 1443208