A characterization of sets in ℝ 2 with DC distance function

Dušan Pokorný; Luděk Zajíček

Czechoslovak Mathematical Journal (2022)

  • Volume: 72, Issue: 1, page 1-38
  • ISSN: 0011-4642

Abstract

top
We give a complete characterization of closed sets F ⊂ ℝ 2 whose distance function d F : = dist ( · , F ) is DC (i.e., is the difference of two convex functions on ℝ 2 ). Using this characterization, a number of properties of such sets is proved.

How to cite

top

Pokorný, Dušan, and Zajíček, Luděk. "A characterization of sets in ${\mathbb {R}}^2$ with DC distance function." Czechoslovak Mathematical Journal 72.1 (2022): 1-38. <http://eudml.org/doc/297914>.

@article{Pokorný2022,
abstract = {We give a complete characterization of closed sets $F \subset \{\mathbb \{R\}\}^2$ whose distance function $d_F:= \{\rm dist\}(\cdot ,F)$ is DC (i.e., is the difference of two convex functions on $\{\mathbb \{R\}\}^2$). Using this characterization, a number of properties of such sets is proved.},
author = {Pokorný, Dušan, Zajíček, Luděk},
journal = {Czechoslovak Mathematical Journal},
keywords = {distance function; DC function; subset of $\{\mathbb \{R\}\}^2$},
language = {eng},
number = {1},
pages = {1-38},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A characterization of sets in $\{\mathbb \{R\}\}^2$ with DC distance function},
url = {http://eudml.org/doc/297914},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Pokorný, Dušan
AU - Zajíček, Luděk
TI - A characterization of sets in ${\mathbb {R}}^2$ with DC distance function
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 1
SP - 1
EP - 38
AB - We give a complete characterization of closed sets $F \subset {\mathbb {R}}^2$ whose distance function $d_F:= {\rm dist}(\cdot ,F)$ is DC (i.e., is the difference of two convex functions on ${\mathbb {R}}^2$). Using this characterization, a number of properties of such sets is proved.
LA - eng
KW - distance function; DC function; subset of ${\mathbb {R}}^2$
UR - http://eudml.org/doc/297914
ER -

References

top
  1. Bačák, M., Borwein, J. M., 10.1080/02331931003770411, Optimization 60 (2011), 961-978. (2011) Zbl1237.46007MR2860286DOI10.1080/02331931003770411
  2. Cannarsa, P., Sinestrari, C., 10.1007/b138356, Progress in Nonlinear Differential Equations and Their Applications 58. Birkhäuser, Boston (2004). (2004) Zbl1095.49003MR2041617DOI10.1007/b138356
  3. Duda, J., 10.1007/s10587-008-0003-1, Czech. Math. J. 58 (2008), 23-49. (2008) Zbl1167.46321MR2402524DOI10.1007/s10587-008-0003-1
  4. Folland, G. B., Real Analysis: Modern Techniques and Their Applications, Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs, and Tracts. John Wiley & Sons, New York (1999). (1999) Zbl0924.28001MR1681462
  5. Fu, J. H. G., Integral geometric regularity, Tensor Valuations and Their Applications in Stochastic Geometry and Imaging Lecture Notes in Mathematics 2177. Springer, Cham (2017), 261-299 9999DOI99999 10.1007/978-3-319-51951-710 . (2017) Zbl1368.53052MR3702376
  6. Fu, J. H. G., Pokorný, D., Rataj, J., 10.1016/j.aim.2017.03.003, Adv. Math. 311 (2017), 796-832. (2017) Zbl1403.53062MR3628231DOI10.1016/j.aim.2017.03.003
  7. Hartman, P., 10.2140/pjm.1959.9.707, Pac. J. Math. 9 (1959), 707-713. (1959) Zbl0093.06401MR0110773DOI10.2140/pjm.1959.9.707
  8. Pokorný, D., Rataj, J., 10.1016/j.aim.2013.08.022, Adv. Math. 248 (2013), 963-985. (2013) Zbl1286.58002MR3107534DOI10.1016/j.aim.2013.08.022
  9. Pokorný, D., Rataj, J., Zajíček, L., 10.1002/mana.201700253, Math. Nachr. 292 (2019), 1595-1626. (2019) Zbl1429.26022MR3982330DOI10.1002/mana.201700253
  10. Pokorný, D., Zajíček, L., 10.1016/j.jmaa.2019.123536, J. Math. Anal. Appl. 482 (2020), Article ID 123536, 14 pages. (2020) Zbl1434.26028MR4015277DOI10.1016/j.jmaa.2019.123536
  11. Pokorný, D., Zajíček, L., 10.14712/1213-7243.2021.006, Commentat. Math. Univ. Carol. 62 (2021), 81-94. (2021) MR4270468DOI10.14712/1213-7243.2021.006
  12. Roberts, A. W., Varberg, D. E., Convex Functions, Pure and Applied Mathematics 57. Academic Press, New York (1973). (1973) Zbl0271.26009MR0442824
  13. Saks, S., Theory of the Integral, Dover Publications, New York (1964). (1964) Zbl1196.28001MR0167578
  14. Shapiro, A., 10.1007/BF00940933, J. Optimization Theory Appl. 66 (1990), 477-487. (1990) Zbl0682.49015MR1080259DOI10.1007/BF00940933
  15. Tuy, H., 10.1007/978-3-319-31484-6, Springer Optimization and Its Applications 110. Springer, Cham (2016). (2016) Zbl1362.90001MR3560830DOI10.1007/978-3-319-31484-6
  16. Veselý, L., Zajíček, L., Delta-convex mappings between Banach spaces and applications, Diss. Math. 289 (1989), 48 pages. (1989) Zbl0685.46027MR1016045
  17. Veselý, L., Zajíček, L., 10.21136/MB.2008.140621, Math. Bohem. 133 (2008), 321-335. (2008) Zbl1199.47242MR2494785DOI10.21136/MB.2008.140621
  18. Willard, S., General Topology, Addison-Wesley Series in Mathematics. Addison-Wesley Publishing, Reading (1970). (1970) Zbl0205.26601MR0264581

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.