Volume approximation of smooth convex bodies by three-polytopes of restricted number of edges
. For a given convex body K in with C 2 boundary, let P c n be the circumscribed polytope of minimal volume with at most n edges, and let P i n be the inscribed polytope of maximal volume with at most n edges. Besides presenting an asymptotic formula for the volume difference as n tends to infinity...
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| Published in | Monatshefte für Mathematik Vol. 153; no. 1; pp. 25 - 48 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Vienna
Springer-Verlag
01.01.2008
|
| Subjects | |
| Online Access | Get full text |
| ISSN | 0026-9255 1436-5081 |
| DOI | 10.1007/s00605-007-0496-y |
Cover
| Abstract | .
For a given convex body
K
in
with
C
2
boundary, let
P
c
n
be the circumscribed polytope of minimal volume with at most
n
edges, and let
P
i
n
be the inscribed polytope of maximal volume with at most
n
edges. Besides presenting an asymptotic formula for the volume difference as
n
tends to infinity in both cases, we prove that the typical faces of
P
c
n
and
P
i
n
are asymptotically regular triangles and squares, respectively, in a suitable sense. |
|---|---|
| AbstractList | .
For a given convex body
K
in
with
C
2
boundary, let
P
c
n
be the circumscribed polytope of minimal volume with at most
n
edges, and let
P
i
n
be the inscribed polytope of maximal volume with at most
n
edges. Besides presenting an asymptotic formula for the volume difference as
n
tends to infinity in both cases, we prove that the typical faces of
P
c
n
and
P
i
n
are asymptotically regular triangles and squares, respectively, in a suitable sense. |
| Author | Tick, Péter Böröczky, Károly J. Gomis, Salvador S. |
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| Keywords | 2000 Mathematics Subject Classification: 52A27, 52A40 Key words: Polytopal approximation, extremal problems |
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| References | Böröczky, Ludwig (CR6) 1999; 127 Bárány (CR2) 2002; 70 Gruber (CR17) 2001; 84 Gruber (CR14) 1993; 5 Schneider (CR25) 1993 Gruber (CR12) 1988; 281 CR13 Leichtweiß (CR21) 1998 Gruber (CR15) 1997; 50 Schneider (CR24) 1987; 128 Bonnesen, Fenchel (CR3) 1987 Fejes Tóth (CR10) 1972 Böröczky (CR4) 2000; 153 Böröczky (CR5) 2000; 102 Fejes Tóth (CR9) 2001; 38 Gruber (CR16) 1998; 10 Gruber (CR18) 2002; 49 CR7 Aurenhammer (CR1) 1987; 16 Schneider (CR23) 1981; 256 CR20 Gruber (CR19) 2004; 186 Glasauer, Gruber (CR11) 1997; 9 Brass, Moser, Pach (CR8) 2005 Ludwig (CR22) 1999; 46 G Fejes Tóth (496_CR9) 2001; 38 R Schneider (496_CR25) 1993 F Aurenhammer (496_CR1) 1987; 16 PM Gruber (496_CR19) 2004; 186 PM Gruber (496_CR15) 1997; 50 PM Gruber (496_CR17) 2001; 84 K Böröczky Jr (496_CR4) 2000; 153 496_CR7 496_CR20 I Bárány (496_CR2) 2002; 70 K Böröczky Jr (496_CR5) 2000; 102 M Ludwig (496_CR22) 1999; 46 T Bonnesen (496_CR3) 1987 K Leichtweiß (496_CR21) 1998 R Schneider (496_CR24) 1987; 128 S Glasauer (496_CR11) 1997; 9 PM Gruber (496_CR16) 1998; 10 PM Gruber (496_CR18) 2002; 49 PM Gruber (496_CR14) 1993; 5 R Schneider (496_CR23) 1981; 256 K Böröczky Jr (496_CR6) 1999; 127 PM Gruber (496_CR12) 1988; 281 P Brass (496_CR8) 2005 L Fejes Tóth (496_CR10) 1972 496_CR13 |
| References_xml | – volume: 153 start-page: 325 year: 2000 end-page: 341 ident: CR4 article-title: Approximation of general smooth convex bodies publication-title: Adv Math doi: 10.1006/aima.1999.1904 – volume: 5 start-page: 281 year: 1993 end-page: 297 ident: CR14 article-title: Asymptotic estimates for best and stepwise approximation of convex bodies I publication-title: Forum Math doi: 10.1515/form.1993.5.281 – volume: 102 start-page: 263 year: 2000 end-page: 285 ident: CR5 article-title: Polytopal approximation bounding the number of -faces publication-title: J Approximation Th doi: 10.1006/jath.1999.3413 – volume: 256 start-page: 289 year: 1981 end-page: 301 ident: CR23 article-title: Zur optimalen Approximation konvexer Hyperflächen durch Polyeder publication-title: Math Ann doi: 10.1007/BF01679698 – volume: 281 start-page: 229 year: 1988 end-page: 245 ident: CR12 article-title: Volume approximation of convex bodies by inscribed polytopes publication-title: Math Ann doi: 10.1007/BF01458430 – volume: 46 start-page: 103 year: 1999 end-page: 125 ident: CR22 article-title: Asymptotic approximation of smooth convex bodies by general polytopes publication-title: Mathematika doi: 10.1112/S0025579300007609 – volume: 186 start-page: 456 year: 2004 end-page: 497 ident: CR19 article-title: Optimum quantization and its applications publication-title: Adv Math doi: 10.1016/j.aim.2003.07.017 – volume: 49 start-page: 227 year: 2002 end-page: 251 ident: CR18 article-title: Optimale Quantisierung publication-title: Math Semesterber – start-page: 1934 year: 1987 ident: CR3 publication-title: Theory of Convex Bodies. 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| Snippet | .
For a given convex body
K
in
with
C
2
boundary, let
P
c
n
be the circumscribed polytope of minimal volume with at most
n
edges, and let
P
i
n
be the... |
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| SubjectTerms | Mathematics Mathematics and Statistics |
| Title | Volume approximation of smooth convex bodies by three-polytopes of restricted number of edges |
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