Nonconvex Lipschitz function in plane which is locally convex outside a discontinuum

Dušan Pokorný

Commentationes Mathematicae Universitatis Carolinae (2014)

  • Volume: 55, Issue: 4, page 509-521
  • ISSN: 0010-2628

Abstract

top
We construct a Lipschitz function on ℝ 2 which is locally convex on the complement of some totally disconnected compact set but not convex. Existence of such function disproves a theorem that appeared in a paper by L. Pasqualini and was also cited by other authors.

How to cite

top

Pokorný, Dušan. "Nonconvex Lipschitz function in plane which is locally convex outside a discontinuum." Commentationes Mathematicae Universitatis Carolinae 55.4 (2014): 509-521. <http://eudml.org/doc/261975>.

@article{Pokorný2014,
abstract = {We construct a Lipschitz function on $\mathbb \{R\}^2$ which is locally convex on the complement of some totally disconnected compact set but not convex. Existence of such function disproves a theorem that appeared in a paper by L. Pasqualini and was also cited by other authors.},
author = {Pokorný, Dušan},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {convex function; convex set; exceptional set; convex function; locally convex function; totally disconnected set; convex set; exceptional set},
language = {eng},
number = {4},
pages = {509-521},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Nonconvex Lipschitz function in plane which is locally convex outside a discontinuum},
url = {http://eudml.org/doc/261975},
volume = {55},
year = {2014},
}

TY - JOUR
AU - Pokorný, Dušan
TI - Nonconvex Lipschitz function in plane which is locally convex outside a discontinuum
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2014
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 55
IS - 4
SP - 509
EP - 521
AB - We construct a Lipschitz function on $\mathbb {R}^2$ which is locally convex on the complement of some totally disconnected compact set but not convex. Existence of such function disproves a theorem that appeared in a paper by L. Pasqualini and was also cited by other authors.
LA - eng
KW - convex function; convex set; exceptional set; convex function; locally convex function; totally disconnected set; convex set; exceptional set
UR - http://eudml.org/doc/261975
ER -

References

top
  1. Burago Ju D., Zalgaller V.A., Sufficient tests for convexity, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 45 (1974), 3–52. MR0377693
  2. Dmitriev V.G., On the construction of ℋ n - 1 -almost everywhere convex hypersurface in ℝ n + 1 , Mat. Sb. (N.S.) 114(156) (1981), 511–522. MR0615339
  3. Kirszbraun M.D., Über die zusammenziehende und Lipschitzsche Transformationen, Fund. Math. 22 (1934), 77–108. 
  4. Pasqualini L., Sur les conditions de convexité d'une variété, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (4) 2 (1938), 1–45. Zbl0026.08801MR1508453

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.