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Floating body, illumination body, and polytopal approximation
C. R. Acad. Sci. Paris, t. 324, S&ie I, p. 201-203, GCom&rie/Geomeffy
Floating body, ill umination and polytopal approximation ..
1997
body,
Carsten SCHUTT
Mathematisches Seminar, Christian Albrechts Universidt, D-24098 Kiel, Germany; Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078, USA.
Abstract.
Let K be a convex body in Wd and Kt its floating bodies. There is a polytope P,, with at most n vertices that satisfies Iit C P,, C I< where Let K1 be the illumination bodies of K and Q,> a polytope that contains K and has at most n d - l-dimensional faces. Then void (li*\Ii) 5 cd4 void (Qn\l<)
Corps jlottant, corps d’illumination et approximation par des polytopes R&urn&
Soient K urn corps convexe dans Rd et K, ses corps flottants. Alors il existe un polytope P,, avec n sornmets tel que ii, c P,, c I,:
Soient K’ les corps d’illumination de K et soit Q,, un polytope qui contient K et qui a n faces de dimension d - 1. Alors on a fold (li’\li) 5 Cd vol,j (Q,,\Ii)
Note pdsentie
par Gilles PISIER.
0764~4442/97/03240201 0 Acadkmie des SciencesElsevier, Paris
201
c. schiitt
Version
franqaise
abrt!g&e
Dans cette Note, nous g6nCralisons corps convexes par des polytopes. Le demi-espaces dont le compltmentaire Lc corps d’illumination h” de K
des rksultats de [ 11 et de [2], [3] sur l’approximation corps flottant Kt d’un corps convexe K est l’intersection coupe K en un ensemble de volume t 141. est [S].
{ .I’ E UP 1v-01,,([CC. K]\K) Tt&oRhIE
jiottunt.
des des
5 t }.
1. - Soient K un corp.~ con\‘exe duns R”, et pour 0 5 t < $ e:-’ ~01,~(K)K, Alors il r.ri.rte un polytope I’,, uvec IL sommets tel que
son corps
K+ c P,, c K
Nous choisissons les sommets .I’~. . . :I:,, E i3K du polytope P, de la man&e suivante. Soit iV( :~:k) une normale $ 3X en :{:A.. On dkigne par H(:r:, I) l’hyperplan contenant :I: et orthogonal B [, et par HP et H + les demi-espaces limit& par H. On choisit .cl arbitrairement, et en supposant choisis .I’]. .I’A.-~ on trouve :I:A. tel que
(.I:~. oti Ak
.c~-~} n Id, (K n H-(.I,~ - &N(:Q.),
N(Y~)))
= fz
est dCtermin& par WA,, (K
n H-(.I:~
- &N(:I:~.).
On ktablit le thCor&me en montrant que cette construction rt de fois. TH~OR~?ME 2. - Snit K un corps convex
N(:f;,,)))
= t.
ne peut &tre rCpCtCe qu’un certain nombre
duns &I” tel yue
1 II; c K c c2B;. Cl
Alors, pour tout polytope I’,, wntencmt K et n ‘avant pas plus de rr jbces de dimension d - 1 on II vol,, (K’\K)
5 lO’rl”(c~~~~)~+~
~1,~ (PT1\K).
Les details de la dtmonstration paraftront prochainement
In this Note we are extending results of I I] and 121, [3] on the approximation of convex bodies by polytopes. The convex floating body Kt of a convex body K is the intersection of all halfspaces whose defining hyperplanes cut off a set of volume f from K [4]. The illumination body K’ of a convex body K is IS] {x E ET’ ) vol,, ([.I:. K]\K)
202
5 t}.
Floating
body,
illumination
body,
and
polytopal
1 - Let K be a convex body in R”. Then we have ,for eve0 (K), that there exist an 71 E N with
THEOREM
+ 6:-3 v&
and a polyope
approximation
t .sati.sfiing 0 5 t 5
P,,, that has 11 vertices and such that K, c P, c K.
We choose the vertices xl, z,, f irK of the palytope P, in the following way. iV(.X$,.) denotes the normal to 3K at
1:~. . , .I:~.-~}n 111t(K n H-(:I:~. - &.N(Q.), N(.c~.))) = 23 where
A,. is determined
by void (K n H-(:rk.
- AeN(:cA.),
N(q.)))
= t.
We claim that this process terminates
for some rt with 4 (K\K,) 71 < P’b(’ f Vd,f (B$)
This claim proves the theorem. Indeed. if we cannot choose another :I:,,+~. then there is no cap of volume t that does not contain an element of the polytope P,:, = [:I:~, . I,,]. By the theorem of Hahn-Banach, we get K, C P,,. 2. - Let K be a convex body in 03” such that
THEOREM
Let 0 < f 5 (5~:1c~)-“~~
Then, ,for every po/yope
w 2< I< - ‘1. - & vdi(K’\W.
vol,, (K)
and /et 71,E N with
P, that contains K and ha,s at most w, d - 1-dirntwsional vol,, (K+\K)
< 10’ r12(qc2)- ‘+A
fures, we have
vol,, (F’?,\K).
The details of the proofs are worked out in a forthcoming paper. This Note was written while the author was visiting the MSRI at Berkeley in the spring of 1996. Note remise le 1” juillet 1996. acceptee le 24 septembre 1996.
References
[2] Dudley 10,
R., 1974.
Metric entropy of’ some classes of \eri wuh differentiableboundaries,Jo~rmrrl
01 Appmximufion
Thror~.
pp. 227-236
[3] Dudley
R., 1979.
Appmximatinn
141 Schiitt [5] Werner
Theory,
Correction 26,
to “Metric entropy of kome 192- 193. E., 1990. The convex floating body.
classes
of
sets
with
differentiable
boundaries”.
Journul
of
pp.
C. and Werner E., 1994. tllumination Bodies
and
the Affine
Surface
Sccmdinuviccr, 66, pp. 275290. StudioMuthemutic~cr, 1 IO. pp. 257-269.
Muthematic~a Area,
203
Floating body, ill umination and polytopal approximation ..
1997
body,
Carsten SCHUTT
Mathematisches Seminar, Christian Albrechts Universidt, D-24098 Kiel, Germany; Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078, USA.
Abstract.
Let K be a convex body in Wd and Kt its floating bodies. There is a polytope P,, with at most n vertices that satisfies Iit C P,, C I< where Let K1 be the illumination bodies of K and Q,> a polytope that contains K and has at most n d - l-dimensional faces. Then void (li*\Ii) 5 cd4 void (Qn\l<)
Corps jlottant, corps d’illumination et approximation par des polytopes R&urn&
Soient K urn corps convexe dans Rd et K, ses corps flottants. Alors il existe un polytope P,, avec n sornmets tel que ii, c P,, c I,:
Soient K’ les corps d’illumination de K et soit Q,, un polytope qui contient K et qui a n faces de dimension d - 1. Alors on a fold (li’\li) 5 Cd vol,j (Q,,\Ii)
Note pdsentie
par Gilles PISIER.
0764~4442/97/03240201 0 Acadkmie des SciencesElsevier, Paris
201
c. schiitt
Version
franqaise
abrt!g&e
Dans cette Note, nous g6nCralisons corps convexes par des polytopes. Le demi-espaces dont le compltmentaire Lc corps d’illumination h” de K
des rksultats de [ 11 et de [2], [3] sur l’approximation corps flottant Kt d’un corps convexe K est l’intersection coupe K en un ensemble de volume t 141. est [S].
{ .I’ E UP 1v-01,,([CC. K]\K) Tt&oRhIE
jiottunt.
des des
5 t }.
1. - Soient K un corp.~ con\‘exe duns R”, et pour 0 5 t < $ e:-’ ~01,~(K)K, Alors il r.ri.rte un polytope I’,, uvec IL sommets tel que
son corps
K+ c P,, c K
Nous choisissons les sommets .I’~. . . :I:,, E i3K du polytope P, de la man&e suivante. Soit iV( :~:k) une normale $ 3X en :{:A.. On dkigne par H(:r:, I) l’hyperplan contenant :I: et orthogonal B [, et par HP et H + les demi-espaces limit& par H. On choisit .cl arbitrairement, et en supposant choisis .I’]. .I’A.-~ on trouve :I:A. tel que
(.I:~. oti Ak
.c~-~} n Id, (K n H-(.I,~ - &N(:Q.),
N(Y~)))
= fz
est dCtermin& par WA,, (K
n H-(.I:~
- &N(:I:~.).
On ktablit le thCor&me en montrant que cette construction rt de fois. TH~OR~?ME 2. - Snit K un corps convex
N(:f;,,)))
= t.
ne peut &tre rCpCtCe qu’un certain nombre
duns &I” tel yue
1 II; c K c c2B;. Cl
Alors, pour tout polytope I’,, wntencmt K et n ‘avant pas plus de rr jbces de dimension d - 1 on II vol,, (K’\K)
5 lO’rl”(c~~~~)~+~
~1,~ (PT1\K).
Les details de la dtmonstration paraftront prochainement
In this Note we are extending results of I I] and 121, [3] on the approximation of convex bodies by polytopes. The convex floating body Kt of a convex body K is the intersection of all halfspaces whose defining hyperplanes cut off a set of volume f from K [4]. The illumination body K’ of a convex body K is IS] {x E ET’ ) vol,, ([.I:. K]\K)
202
5 t}.
Floating
body,
illumination
body,
and
polytopal
1 - Let K be a convex body in R”. Then we have ,for eve0 (K), that there exist an 71 E N with
THEOREM
+ 6:-3 v&
and a polyope
approximation
t .sati.sfiing 0 5 t 5
P,,, that has 11 vertices and such that K, c P, c K.
We choose the vertices xl, z,, f irK of the palytope P, in the following way. iV(.X$,.) denotes the normal to 3K at
1:~. . , .I:~.-~}n 111t(K n H-(:I:~. - &.N(Q.), N(.c~.))) = 23 where
A,. is determined
by void (K n H-(:rk.
- AeN(:cA.),
N(q.)))
= t.
We claim that this process terminates
for some rt with 4 (K\K,) 71 < P’b(’ f Vd,f (B$)
This claim proves the theorem. Indeed. if we cannot choose another :I:,,+~. then there is no cap of volume t that does not contain an element of the polytope P,:, = [:I:~, . I,,]. By the theorem of Hahn-Banach, we get K, C P,,. 2. - Let K be a convex body in 03” such that
THEOREM
Let 0 < f 5 (5~:1c~)-“~~
Then, ,for every po/yope
w 2< I< - ‘1. - & vdi(K’\W.
vol,, (K)
and /et 71,E N with
P, that contains K and ha,s at most w, d - 1-dirntwsional vol,, (K+\K)
< 10’ r12(qc2)- ‘+A
fures, we have
vol,, (F’?,\K).
The details of the proofs are worked out in a forthcoming paper. This Note was written while the author was visiting the MSRI at Berkeley in the spring of 1996. Note remise le 1” juillet 1996. acceptee le 24 septembre 1996.
References
[2] Dudley 10,
R., 1974.
Metric entropy of’ some classes of \eri wuh differentiableboundaries,Jo~rmrrl
01 Appmximufion
Thror~.
pp. 227-236
[3] Dudley
R., 1979.
Appmximatinn
141 Schiitt [5] Werner
Theory,
Correction 26,
to “Metric entropy of kome 192- 193. E., 1990. The convex floating body.
classes
of
sets
with
differentiable
boundaries”.
Journul
of
pp.
C. and Werner E., 1994. tllumination Bodies
and
the Affine
Surface
Sccmdinuviccr, 66, pp. 275290. StudioMuthemutic~cr, 1 IO. pp. 257-269.
Muthematic~a Area,
203