| Title:
|
Nonconvex Lipschitz function in plane which is locally convex outside a discontinuum (English) |
| Author:
|
Pokorný, Dušan |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
55 |
| Issue:
|
4 |
| Year:
|
2014 |
| Pages:
|
509-521 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
convex function |
| Keyword:
|
convex set |
| Keyword:
|
exceptional set |
| MSC:
|
26B25 |
| MSC:
|
52A20 |
| MSC:
|
52A41 |
| idZBL:
|
Zbl 06391559 |
| idMR:
|
MR3269013 |
| . |
| Date available:
|
2014-10-09T09:57:48Z |
| Last updated:
|
2017-01-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143975 |
| . |
| Reference:
|
[1] Burago Ju D., Zalgaller V.A.: Sufficient tests for convexity.Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 45 (1974), 3–52. MR 0377693 |
| Reference:
|
[2] Dmitriev V.G.: On the construction of $\mathcal H_{n-1}$-almost everywhere convex hypersurface in $\mathbb R^{n+1}$.Mat. Sb. (N.S.) 114(156) (1981), 511–522. MR 0615339 |
| Reference:
|
[3] Kirszbraun M.D.: Über die zusammenziehende und Lipschitzsche Transformationen.Fund. Math. 22 (1934), 77–108. |
| Reference:
|
[4] Pasqualini L.: Sur les conditions de convexité d'une variété.Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (4) 2 (1938), 1–45. Zbl 0026.08801, MR 1508453 |
| . |
