Curvature bounds for neighborhoods of self-similar sets

Steffen Winter

Commentationes Mathematicae Universitatis Carolinae (2011)

  • Volume: 52, Issue: 2, page 205-226
  • ISSN: 0010-2628

Abstract

top
In some recent work, fractal curvatures C k f ( F ) and fractal curvature measures C k f ( F , · ) , k = 0 , ... , d , have been determined for all self-similar sets F in ℝ d , for which the parallel neighborhoods satisfy a certain regularity condition and a certain rather technical curvature bound. The regularity condition is conjectured to be always satisfied, while the curvature bound has recently been shown to fail in some concrete examples. As a step towards a better understanding of its meaning, we discuss several equivalent formulations of the curvature bound condition and also a very natural technically simpler condition which turns out to be stronger. These reformulations show that the validity of this condition does not depend on the choice of the open set and the constant R appearing in the condition and allow to discuss some concrete examples of self-similar sets. In particular, it is shown that the class of sets satisfying the curvature bound condition is strictly larger than the class of sets satisfying the assumption of polyconvexity used in earlier results.

How to cite

top

Winter, Steffen. "Curvature bounds for neighborhoods of self-similar sets." Commentationes Mathematicae Universitatis Carolinae 52.2 (2011): 205-226. <http://eudml.org/doc/247023>.

@article{Winter2011,
abstract = {In some recent work, fractal curvatures $C^f_k(F)$ and fractal curvature measures $C^f_k(F,\cdot )$, $k= 0,\ldots ,d$, have been determined for all self-similar sets $F$ in $\mathbb \{R\}^d$, for which the parallel neighborhoods satisfy a certain regularity condition and a certain rather technical curvature bound. The regularity condition is conjectured to be always satisfied, while the curvature bound has recently been shown to fail in some concrete examples. As a step towards a better understanding of its meaning, we discuss several equivalent formulations of the curvature bound condition and also a very natural technically simpler condition which turns out to be stronger. These reformulations show that the validity of this condition does not depend on the choice of the open set and the constant $R$ appearing in the condition and allow to discuss some concrete examples of self-similar sets. In particular, it is shown that the class of sets satisfying the curvature bound condition is strictly larger than the class of sets satisfying the assumption of polyconvexity used in earlier results.},
author = {Winter, Steffen},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {self-similar set; parallel set; curvature measures; fractal curvatures; Minkowski content; Minkowski dimension; regularity condition; curvature bound condition; self-similar set; parallel set; curvature measure; fractal curvatures; Minkowski content; Minkowski dimension; regularity condition; curvature bound condition},
language = {eng},
number = {2},
pages = {205-226},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Curvature bounds for neighborhoods of self-similar sets},
url = {http://eudml.org/doc/247023},
volume = {52},
year = {2011},
}

TY - JOUR
AU - Winter, Steffen
TI - Curvature bounds for neighborhoods of self-similar sets
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2011
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 52
IS - 2
SP - 205
EP - 226
AB - In some recent work, fractal curvatures $C^f_k(F)$ and fractal curvature measures $C^f_k(F,\cdot )$, $k= 0,\ldots ,d$, have been determined for all self-similar sets $F$ in $\mathbb {R}^d$, for which the parallel neighborhoods satisfy a certain regularity condition and a certain rather technical curvature bound. The regularity condition is conjectured to be always satisfied, while the curvature bound has recently been shown to fail in some concrete examples. As a step towards a better understanding of its meaning, we discuss several equivalent formulations of the curvature bound condition and also a very natural technically simpler condition which turns out to be stronger. These reformulations show that the validity of this condition does not depend on the choice of the open set and the constant $R$ appearing in the condition and allow to discuss some concrete examples of self-similar sets. In particular, it is shown that the class of sets satisfying the curvature bound condition is strictly larger than the class of sets satisfying the assumption of polyconvexity used in earlier results.
LA - eng
KW - self-similar set; parallel set; curvature measures; fractal curvatures; Minkowski content; Minkowski dimension; regularity condition; curvature bound condition; self-similar set; parallel set; curvature measure; fractal curvatures; Minkowski content; Minkowski dimension; regularity condition; curvature bound condition
UR - http://eudml.org/doc/247023
ER -

References

top
  1. Cheeger J., Müller W., Schrader R., 10.1007/BF01210729, Comm. Math. Phys. 92 (1984), 405–454. MR0734226DOI10.1007/BF01210729
  2. Bröcker L., Kuppe M., 10.1023/A:1005248711077, Geom. Dedicata 82 (2000), 1897–1924. MR1789065DOI10.1023/A:1005248711077
  3. Falconer K.J., 10.1090/S0002-9939-1995-1224615-4, Proc. Am. Math. Soc. 123 (1995), no. 4, 1115–1124. Zbl0838.28006MR1224615DOI10.1090/S0002-9939-1995-1224615-4
  4. Federer H., 10.1090/S0002-9947-1959-0110078-1, Trans. Amer. Math. Soc. 93 (1959), 418–491. Zbl0089.38402MR0110078DOI10.1090/S0002-9947-1959-0110078-1
  5. Fu J.H.G., 10.1215/S0012-7094-85-05254-8, Duke Math. J. 52 (1985), 1025–1046. Zbl0592.52002MR0816398DOI10.1215/S0012-7094-85-05254-8
  6. Gatzouras D., 10.1090/S0002-9947-99-02539-8, Trans. Amer. Math. Soc. 352 (2000), no. 5, 1953–1983. Zbl0946.28006MR1694290DOI10.1090/S0002-9947-99-02539-8
  7. Hutchinson J.E., 10.1512/iumj.1981.30.30055, Indiana Univ. Math. J. 30 (1981), 713–747. Zbl0598.28011MR0625600DOI10.1512/iumj.1981.30.30055
  8. Lapidus M.L., Pomerance C., The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums, Proc. London Math. Soc. 66 (1993), no. 1, 41–69. Zbl0739.34065MR1189091
  9. Llorente M., Winter S., 10.1002/mana.200410471, Math. Nachr. 280 (2007), no. 1–2, 152–170. Zbl1118.28006MR2290389DOI10.1002/mana.200410471
  10. Rataj J., Winter S., On volume and surface area of parallel sets, Indiana Univ. Math. J., to appear, online-preprint: http://www.iumj.indiana.edu/IUMJ/Preprints/4165.pdf. 
  11. Schief A., 10.1090/S0002-9939-1994-1191872-1, Proc. Amer. Math. Soc. 122 (1994), no. 1, 111–115. Zbl0807.28005MR1191872DOI10.1090/S0002-9939-1994-1191872-1
  12. Winter S., 10.4064/dm453-0-1, Dissertationes Math. 453 (2008), 1–66. Zbl1139.28300MR2423952DOI10.4064/dm453-0-1
  13. Winter S., Zähle M., Fractal curvature measures of self-similar sets, submitted, arxiv.org/abs/1007.0696v2. 
  14. Zähle M., 10.1007/BF00181273, Geom. Dedicata 23 (1987), 155–171. MR0892398DOI10.1007/BF00181273
  15. Zähle M., 10.1007/BF00127938, Ann. Global Anal. Geom. 8 (1990), 249–260. MR1089237DOI10.1007/BF00127938
  16. Zähle M., 10.1090/S0002-9947-2010-05198-0, Trans. Amer. Math. Soc. 363 (2011), 2663–2684. MR2763731DOI10.1090/S0002-9947-2010-05198-0

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.