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The length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083920/s0839201.png" /> of a centrally-symmetric closed convex curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083920/s0839202.png" /> on a plane, measured in the Minkowski metric for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083920/s0839203.png" /> plays the role of unit circle. Always <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083920/s0839204.png" /> (see [[#References|[1]]]). In the generalization to the non-symmetric case, a self-perimeter is the length of a directed curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083920/s0839205.png" />, and depends on the choice of the origin inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083920/s0839206.png" /> (see [[#References|[2]]], [[#References|[3]]]). The case of a star <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083920/s0839207.png" /> has been considered (see [[#References|[4]]]). There are various generalizations of a self-perimeter for the unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083920/s0839208.png" /> in a normed space of dimension greater than two (see [[#References|[5]]], [[#References|[6]]]).
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The length $2\pi_S$ of a centrally-symmetric closed convex curve $S$ on a plane, measured in the Minkowski metric for which $S$ plays the role of unit circle. Always $3\leq\pi_S\leq4$ (see [[#References|[1]]]). In the generalization to the non-symmetric case, a self-perimeter is the length of a directed curve $S$, and depends on the choice of the origin inside $S$ (see [[#References|[2]]], [[#References|[3]]]). The case of a star $S$ has been considered (see [[#References|[4]]]). There are various generalizations of a self-perimeter for the unit sphere $S$ in a normed space of dimension greater than two (see [[#References|[5]]], [[#References|[6]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.G. Reshetnyak,  "An extremal problem in the theory of surfaces"  ''Uspekhi Mat. Nauk'' , '''8''' :  6  (1953)  pp. 125–126  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.A. Sorokin,  "Minkowski geometry with asymmetric indicatrix"  ''Uchebn. Zap. Moskov. Gos. Ped. Inst.'' , '''243'''  (1965)  pp. 160–185  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.D. Chakerian,  W.K. Talley,  "Some properties of the self-circumference of convex sets"  ''Arch. Math.'' , '''20''' :  4  (1969)  pp. 431–443</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Gołab,  "Sur la longuer d'indicatrice dans la géometrie plane de Minkowski"  ''Colloq. Math.'' , '''15''' :  1  (1966)  pp. 141–144</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.J. Schäffer,  "Spheres with maximum inner diameter"  ''Math. Ann.'' , '''190''' :  3  (1971)  pp. 242–247</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  C.M. Petty,  "Geominimal surface area"  ''Geom. Dedic.'' , '''3''' :  1  (1974)  pp. 77–97</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.G. Reshetnyak,  "An extremal problem in the theory of surfaces"  ''Uspekhi Mat. Nauk'' , '''8''' :  6  (1953)  pp. 125–126  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.A. Sorokin,  "Minkowski geometry with asymmetric indicatrix"  ''Uchebn. Zap. Moskov. Gos. Ped. Inst.'' , '''243'''  (1965)  pp. 160–185  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.D. Chakerian,  W.K. Talley,  "Some properties of the self-circumference of convex sets"  ''Arch. Math.'' , '''20''' :  4  (1969)  pp. 431–443</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Gołab,  "Sur la longuer d'indicatrice dans la géometrie plane de Minkowski"  ''Colloq. Math.'' , '''15''' :  1  (1966)  pp. 141–144</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.J. Schäffer,  "Spheres with maximum inner diameter"  ''Math. Ann.'' , '''190''' :  3  (1971)  pp. 242–247</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  C.M. Petty,  "Geominimal surface area"  ''Geom. Dedic.'' , '''3''' :  1  (1974)  pp. 77–97</TD></TR></table>

Revision as of 14:01, 1 October 2014

The length $2\pi_S$ of a centrally-symmetric closed convex curve $S$ on a plane, measured in the Minkowski metric for which $S$ plays the role of unit circle. Always $3\leq\pi_S\leq4$ (see [1]). In the generalization to the non-symmetric case, a self-perimeter is the length of a directed curve $S$, and depends on the choice of the origin inside $S$ (see [2], [3]). The case of a star $S$ has been considered (see [4]). There are various generalizations of a self-perimeter for the unit sphere $S$ in a normed space of dimension greater than two (see [5], [6]).

References

[1] Yu.G. Reshetnyak, "An extremal problem in the theory of surfaces" Uspekhi Mat. Nauk , 8 : 6 (1953) pp. 125–126 (In Russian)
[2] V.A. Sorokin, "Minkowski geometry with asymmetric indicatrix" Uchebn. Zap. Moskov. Gos. Ped. Inst. , 243 (1965) pp. 160–185 (In Russian)
[3] G.D. Chakerian, W.K. Talley, "Some properties of the self-circumference of convex sets" Arch. Math. , 20 : 4 (1969) pp. 431–443
[4] S. Gołab, "Sur la longuer d'indicatrice dans la géometrie plane de Minkowski" Colloq. Math. , 15 : 1 (1966) pp. 141–144
[5] J.J. Schäffer, "Spheres with maximum inner diameter" Math. Ann. , 190 : 3 (1971) pp. 242–247
[6] C.M. Petty, "Geominimal surface area" Geom. Dedic. , 3 : 1 (1974) pp. 77–97
How to Cite This Entry:
Self-perimeter. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Self-perimeter&oldid=33455
This article was adapted from an original article by V.A. Zalgaller (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article