Selfishness and Rupert Property of convex bodies Liping Yuan College of Mathematics and Information Science Hebei Normal University Shijiazhuang, China Shanghai Jiaotong University
Part I. Selfishness of Convex Bodies Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 2 / 67
F -convexity Tudor Zamfirescu proposed at the 1974 meeting on Convexity in Oberwolfach the investigation of F -convexity, for various families F . Let F be a family of sets in R d . A set M ⊂ R d is called F -convex 1 if for any pair of distinct points x, y ∈ M there is a set F ∈ F such that x, y ∈ F and F ⊂ M . M x F y F ∈ F and x, y ∈ F ⊂ M 1 Blind R, Valette G, Zamfirescu T. Rectangular convexity [J]. Geom. Dedicata, 1980, 9: 317-327. Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 3 / 67
F -convexity members of F the usual convexity usual closed line-segments affine linearity lines arc-wise connectedness arcs polygonal connectedness polygonal paths Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 4 / 67
F is a family of connected sets Authors Members of F F -convexity Blind 2 and B¨ oczky 3 or¨ r -convexity all non-degenerate rectangles all polygonal paths in the plane Bruckner 4 and Magazanik 5 L n -convexity with length at most n 2 Blind R, Valette G, Zamfirescu T. Rectangular convexity [J]. Geom. Dedicata, 1980, 9: 317-327. 3 B¨ or¨ oczky K Jr. Rectangular convexity of convex domains of constant width [J]. Geom. Dedicata, 1990, 34: 13-18. 4 Bruckner A M, Bruckner J B. Generalized convex kernels[J]. Israel J. Math., 1964, 2(1): 27-32. 5 Magazanik E, Perles M A. Generalized convex kernels of simply connected L n sets in the plane[J]. Israel J. Math., 2007, 160(1): 157-171. Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 5 / 67
F is a family of connected sets Authors Members of F F -convexity all polygonal paths in R 2 Magazanik et al. 6 staircase convexity with sides parallel to the coordinate axis right triangles in a Hilbert Zamfirescu 7 right convexity space of dimension at least 2 6 Magazanik E, Perles M A. Staircase connected sets[J]. Discrete Comp. Geom., 2007, 37: 587-599. 7 Zamfirescu T. Right convexity [J]. J. Convex Anal., 2014, 21: 253-260. Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 6 / 67
F is a family of discrete point sets Authors Members of F F -convexity Yuan et al. 8 9 rt -convexity all triples { x, y, z } with ∠ xyz = π 2 all triples whose convex hull Yuan et al. 10 it -convexity are isosceles triangles (maybe degenerated triangles) all quadruple whose convex hull Li et al. 11 rq -convexity is a non-degenerate rectangle 8 Yuan L, Zamfirescu T. Right triple convex completion [J]. J. Convex Anal., 2015, 22(1): 291-301. 9 Yuan L, Zamfirescu T. Right triple convexity [J]. J. Convex Anal., 2016, 23: 1219-1246. 10 Yuan L, Zamfirescu T, Zhang Y. Isosceles triple convexity [J]. Carpathian J. Math., 2017, 33(1): 127-139. 11 Li D, Yuan L, Zamfirescu T. Right quadruple convexity [J]. Ars Math. Contemp., 2018, 14: 25-38. Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 7 / 67
For a subset of the vertex set of a graph, the g -convexity investigated by Farber and Jamison 12 , the T -convexity studied by Changat and Mathew 13 , and M -convexity researched by Duchet 14 can also be regarded as examples of F -convexity for suitable families F . 12 M. Farber, R. E. Jamison. On local convexity in graphs[J]. Discrete Math. , 66 (1987) 231-247. 13 M. Changat, J. Mathew. On triangle path convexity in graphs, Discrete Math ., 206 (1999) 91-95. 14 P. Duchet. Convex sets in graph II. Minimal path convexity, Combin. Theory Ser. B , 44 (1988) 307-316. Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 8 / 67
Selfishness of compact sets A family F of compact sets is complete if F contains all compact F -convex sets. Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 9 / 67
Selfishness of compact sets A family F of compact sets is complete if F contains all compact F -convex sets. A compact set K is called selfish, if the family F K of all compact sets similar to K is complete. Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 9 / 67
Example: Every square is selfish Let Q be a square, and let K be a compact F Q -convex set. We show that K is also a square. Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 10 / 67
Notation As usual, for M ⊂ R d with d ≥ 2 , cl M denotes its topological closure, bd M its boundary, and diam M = sup x,y ∈ M � x − y � . A 2-point set { x, y } ⊂ M with � x − y � = diam M is called a diametral pair of M , while xy is a diameter of M . For any compact set C ⊂ R d , let S C be the smallest hypersphere containing C in its convex hull. Let K be the space of all convex bodies in R d , endowed with the Pompeiu-Hausdorff metric. A convex body K is called long if card ( K ∩ S K ) = 2 . The unit ball is denoted by B , and bd B = S . Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 11 / 67
Non-selfish convex bodies Unimodal A continuous real function defined on an interval [ a, b ] is unimodal if it is non-decreasing on a subinterval [ a, c ] and non-increasing on [ c, b ] . A C 2 -arc A in R 2 will be called here unimodal, if its curvature radius at x ∈ A is a unimodal function of arc-length (from an endpoint of A to x ). Theorem: Suppose K ⊂ R 2 is a long convex body. If at least one of the arcs in bd K between the two points of K ∩ S K is unimodal, then K is not selfish. Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 12 / 67
The condition card( K ∩ S K ) = 2 alone does not guarantee non-selfishness. Not every polygon is selfish either. For instance, if an edge ab of the polygon P is a diameter of S P , then the disc is F P -convex, as one easily verifies. So, among the triangles, all non-acute ones are non-selfish. Are all acute ones selfish? Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 13 / 67
Selfishness of triangles Theorem: The equilateral triangle is selfish. Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 14 / 67
Is every acute triangle is selfish ? Theorem: There exist non-selfish acute triangles. We prove that the triangle x 0 dz 0 is not selfish, by showing that abcde is F x 0 dz 0 -convex. Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 15 / 67
Selfishness of triangles Theorem: Every isosceles acute triangle is selfish. � � �� �� � � � � Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 16 / 67
Selfishness of quadrilaterals Regarding families F of sets, an interesting case is that of all rectangles. The family F is not complete, but a characterization of F -convexity in the compact case is still missing. Theorem: Every rectangle is selfish in the plane. �� � � � �� �� � � � �� Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 17 / 67
Selfishness of quadrilaterals In the case of the rhombus, while every rhombus is selfish, the family R of all of them is not complete, as again the disc demonstrates. Theorem: Every rhombus in R 2 is selfish. Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 18 / 67
Selfishness of regular convex polygons Theorem: Every regular convex polygon is selfish. �� � � � � � � � �� � �� � � � � � � � � � � � � � �� � �� � �� � � � � � � � � � �� � Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 19 / 67
Other non-selfish convex bodies Concerning the selfishness of long polytopes, it is clear that a necessary condition is that their diameter is not contained in a face. Is this also sufficient? We have seen, indeed, that every rhombus is selfish. Theorem: There exists a long polygon with its diameter not among the sides, which is not selfish. Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 20 / 67
Other non-selfish convex bodies As all rectangles are selfish in R 2 , we might be inclined to think that also (some) circular cylinders in R 3 are selfish... Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 21 / 67
Other non-selfish convex bodies As all rectangles are selfish in R 2 , we might be inclined to think that also (some) circular cylinders in R 3 are selfish... Theorem: No circular cylinder in R 3 is selfish. Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 21 / 67
Families of convex bodies and selfishness The convex bodies of constant width are selfish and the family of all of them complete. Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 22 / 67
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