Poidge-Convexity

Bo Wang; Liping Yuan; Tudor Zamfirescu

doi:10.68381/jca28067

Abstract

We investigate in this paper the poidge-convexity, which is a generalization of right convexity, introduced by one of the present authors in 2014. Although not every convex body is poidge-convex, there are many families of compact sets, some of them very different from convex bodies, which are poidge-convex. We present here several such families.

Author information

Contact details are reproduced from the original publication and may be historical.

Bo Wang

School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, P. R. China
and: Hebei Key Laboratory of Computational Mathematics, Shijiazhuang, P. R. China

bow6666@163.com

Liping Yuan

School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, P. R. China
and: Hebei Key Laboratory of Comp. Mathematics and Applications, Shijiazhuang, P. R. China

lpyuan@hebtu.edu.cn

Tudor Zamfirescu

Fachbereich Mathematik, Universität Dortmund, Germany
and: Romanian Academy, Bucharest, Romania
and: School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, P.R.China

tuzamfirescu@gmail.com

Keywords

Poidge-convexity
starshaped sets
topological discs
polyhedral surfaces

Mathematics Subject Classification

52A01, 52A37

References

R. Blind, G. Valette, T. Zamfirescu: Rectangular convexity, Geom. Dedicata 9 (1980) 317–327.https://doi.org/10.1007/BF00181177 Google Scholar
K. Böröczky, Jr.: Rectangular convexity of convex domains of constant width, Geom. Dedicata 34 (1990) 13–18.https://doi.org/10.1007/BF00150684 Google Scholar
M. Breen: Characterizing certain staircase convex sets in Rn, Beiträge Algebra Geometrie 51 (2010) 251–261.Google Scholar
M. Breen: Generating a staircase starshaped set and its kernel in R3 from certain staircase convex subsets, Beiträge Algebra Geometrie 53 (2012) 515–528.https://doi.org/10.1007/s13366-011-0071-5 Google Scholar
M. Breen: Suitable families of boxes and kernels of staircase starshaped sets in Rn, Aequationes Math. 87 (2014) 43–52.https://doi.org/10.1007/s00010-013-0236-4 Google Scholar
M. Breen: Staircase convex results for a certain class of orthogonal polytopes, Aequationes Math. 89 (2015) 1213–1221.https://doi.org/10.1007/s00010-014-0294-2 Google Scholar
A. M. Bruckner, J. B. Bruckner: Generalized convex kernels, Israel J. Math. 2 (1964) 27–32.https://doi.org/10.1007/BF02759731 Google Scholar
H. Buseman: Convex Surfaces, Interscience, New York (1958).Google Scholar
V. Klee: Some new results on smoothness and rotundity in normed linear spaces, Math. Annalen 139 (1959) 51–63.https://doi.org/10.1007/BF01459822 Google Scholar
E. Magazanik, M. A. Perles: Staircase connected sets, Discrete Comp. Geometry 37 (2007) 587–599.https://doi.org/10.1007/s00454-007-1308-9 Google Scholar
E. Magazanik, M. A. Perles: Generalized convex kernels of simply connected Ln sets in the plane, Israel J. Math. 160 (2007) 157–171.https://doi.org/10.1007/s11856-007-0059-x Google Scholar
L. Yuan, T. Zamfirescu: Right triple convex completion, J. Convex Analysis 22 (2015) 291–301.https://doi.org/10.68381/jca22014 Google Scholar
L. Yuan, T. Zamfirescu: Right triple convexity, J. Convex Analysis 23 (2016) 1219–1246.https://doi.org/10.68381/jca23046 Google Scholar
L. Yuan, T. Zamfirescu: Selfishness of convex bodies, J. Nonlinear Convex Analysis 18 (2017) 525–534.Google Scholar
L. Yuan, T. Zamfirescu, Y. Zhang: Isosceles triple convexity, Carpathian J. Math. 33 (2017) 127–139.https://doi.org/10.37193/CJM.2017.01.13 Google Scholar
T. Zamfirescu: Inscribed and circumscribed circles to convex curves, Proc. Amer. Math. Soc. 80 (1980) 455–457.https://doi.org/10.1090/S0002-9939-1980-0581004-2 Google Scholar
T. Zamfirescu: Typical starshaped sets, Aequationes Math. 36 (1988) 188–200.https://doi.org/10.1007/BF01836090 Google Scholar
T. Zamfirescu: Right convexity, J. Convex Analysis 21 (2014) 253–260.https://doi.org/10.68381/jca21014 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2021. Rights now held by Banach Press LLC.

[1] R. Blind, G. Valette, T. Zamfirescu: Rectangular convexity, Geom. Dedicata 9 (1980) 317–327.https://doi.org/10.1007/BF00181177 Google Scholar

[2] K. Böröczky, Jr.: Rectangular convexity of convex domains of constant width, Geom. Dedicata 34 (1990) 13–18.https://doi.org/10.1007/BF00150684 Google Scholar

[3] M. Breen: Characterizing certain staircase convex sets in Rn, Beiträge Algebra Geometrie 51 (2010) 251–261.Google Scholar

[4] M. Breen: Generating a staircase starshaped set and its kernel in R3 from certain staircase convex subsets, Beiträge Algebra Geometrie 53 (2012) 515–528.https://doi.org/10.1007/s13366-011-0071-5 Google Scholar

[5] M. Breen: Suitable families of boxes and kernels of staircase starshaped sets in Rn, Aequationes Math. 87 (2014) 43–52.https://doi.org/10.1007/s00010-013-0236-4 Google Scholar

[6] M. Breen: Staircase convex results for a certain class of orthogonal polytopes, Aequationes Math. 89 (2015) 1213–1221.https://doi.org/10.1007/s00010-014-0294-2 Google Scholar

[7] A. M. Bruckner, J. B. Bruckner: Generalized convex kernels, Israel J. Math. 2 (1964) 27–32.https://doi.org/10.1007/BF02759731 Google Scholar

[8] H. Buseman: Convex Surfaces, Interscience, New York (1958).Google Scholar

[9] V. Klee: Some new results on smoothness and rotundity in normed linear spaces, Math. Annalen 139 (1959) 51–63.https://doi.org/10.1007/BF01459822 Google Scholar

[10] E. Magazanik, M. A. Perles: Staircase connected sets, Discrete Comp. Geometry 37 (2007) 587–599.https://doi.org/10.1007/s00454-007-1308-9 Google Scholar

[11] E. Magazanik, M. A. Perles: Generalized convex kernels of simply connected Ln sets in the plane, Israel J. Math. 160 (2007) 157–171.https://doi.org/10.1007/s11856-007-0059-x Google Scholar

[12] L. Yuan, T. Zamfirescu: Right triple convex completion, J. Convex Analysis 22 (2015) 291–301.https://doi.org/10.68381/jca22014 Google Scholar

[13] L. Yuan, T. Zamfirescu: Right triple convexity, J. Convex Analysis 23 (2016) 1219–1246.https://doi.org/10.68381/jca23046 Google Scholar

[14] L. Yuan, T. Zamfirescu: Selfishness of convex bodies, J. Nonlinear Convex Analysis 18 (2017) 525–534.Google Scholar

[15] L. Yuan, T. Zamfirescu, Y. Zhang: Isosceles triple convexity, Carpathian J. Math. 33 (2017) 127–139.https://doi.org/10.37193/CJM.2017.01.13 Google Scholar

[16] T. Zamfirescu: Inscribed and circumscribed circles to convex curves, Proc. Amer. Math. Soc. 80 (1980) 455–457.https://doi.org/10.1090/S0002-9939-1980-0581004-2 Google Scholar

[17] T. Zamfirescu: Typical starshaped sets, Aequationes Math. 36 (1988) 188–200.https://doi.org/10.1007/BF01836090 Google Scholar

[18] T. Zamfirescu: Right convexity, J. Convex Analysis 21 (2014) 253–260.https://doi.org/10.68381/jca21014 Google Scholar