Hunting for reduced polytopes

0.16 MB·

by  Bernardo González Merino

Hunting for reduced polytopes
Bernardo González Merino † Thomas Jahn‡
∗
Alexandr Polyanskii§¶ Gerd Wachsmuth
k
January 31, 2017
7
1
0
2

n Weshowthatthereexistreducedpolytopesinthree-dimensionalEuclidean
a space. This partiallyanswers the question posed by Lassak [10] on the exis-
J
tenceofreducedpolytopesin d-dimensionalEuclideanspacefor d 3. More-
0 ≥
3 over, we prove a novel necessary condition on reduced polytopes in three-
dimensionalEuclideanspace.

]
G
Keywords: polytope,reducedness
M
MSC(2010): 52B10
.
h
t
a 1 Introduction
m
[
Constantwidthbodies,i.e.,convexbodiesforwhichparallelsupportinghyperplaneshave

1 constantdistance,havealongandrichhistoryinmathematics[5]. DuetoMeissner[20],
v
constant width bodies in Euclidean space can be characterized by diametrical complete-
9
2 ness, that is, the property of not being properly contained in a set of the same diameter.
6 Constant widthbodies also belong to a related class of reducedconvex bodies introduced
8
byHeil[8]. Thismeansthatconstantwidthbodiesdonotproperlycontainaconvexbody
0
. of same minimum width. Remarkably, the classes of reduced bodies and constant width
1
0 bodiesdonotcoincide,asaregulartriangleintheEuclideanplaneshows.
7
1
v: ∗TUMunich,ZentrumMathematik,[email protected]
†ThefirstauthorispartiallysupportedbyConsejeríadeIndustria, Turismo, EmpresaeInnovacióndela
i
X CARMthroughFundaciónSéneca,AgenciadeCienciayTecnologíadelaRegióndeMurcia,Programade
r FormaciónPostdoctoraldePersonalInvestigador19769/PD/15andproject19901/GERM/15,Programme
a inSupportofExcellenceGroupsoftheRegióndeMurcia,andbyMINECOprojectreferenceMTM2015-
63699-P,Spain.
‡TUChemnitz,FacultyofMathematics,[email protected]
§MoscowInstitute ofPhysics andTechnology, Technion, Institute forInformationTransmissionProblems
RAS,[email protected]
¶TheauthorwaspartiallysupportedbytheRussianFoundationforBasicResearch,grantsNo.15-31-20403
(mol_a_ved),No.15-01-99563A,No.15-01-03530A.
TUChemnitz,FacultyofMathematics,[email protected]
k
1
Reduced bodies are extremal in remarkable inequalities for prescribed minimum width,
asinSteinhagen’sinequality[5](minimuminradius),orothersthatsurprisinglystillre-
mainunsolved,namely,Pál’sproblem[21](minimumvolume). Whiletheregularsimplex
(and any of its reduced subsets) is extremal for Steinhagen’s, it is extremal only in the
planar case for Pál’s problem. The reason is that while the regular triangle is reduced,
thisisnolongerthecasefortheregularsimplexinRd,d 3. Indeed,Heilconjectured[8]
≥
that a certain reduced subset of the regular simplex is extremal for Pál’s problem. Heil
also observed that some reduced body has to be extreme for Pál’s problem when replac-
ing volume by quermassintegral. The existence of reduced polytopes, and the fact that
smooth reduced sets are of constant width (cf. [8]), opens the door to conjecture some of
themasminimizers. Infullgenerality,anynon-decreasing-inclusionfunctionalofconvex
bodieswithprescribedminimumwidth,attainsitsminimumatsomereducedbody. Pál’s
problemrestrictedtoconstantwidthsetsisthewell-knownBlaschke–Lebesgueproblem,
cf.[5],solvedonlyintheplanarcase,wheretheReuleauxtriangleistheminimizerofthe
area,andMeissner’sbodiesareconjecturedtobeextremalinthethree-dimensionalspace,
see [15, pp. 216, 217] for an extended discussion. Note that Pál’s problem has also been
investigated in other geometrical settings such as Minkowskian planes [1] or spherical
geometry, cf.[4,pp.96,97]and[17].
Reduced bodies in the Euclidean space have been extensively studied in [10,11,15], and
the concept of reducednesshas been translated to finite-dimensional normed spaces [13,
14,16]. In reference to the existence of reducedpolygons in the Euclidean plane, Lassak
[10] posed the question whether there exist reduced polytopes in Euclidean d-space for
d 3. Several authors addressed the search for reduced polytopes in finite-dimensional
≥
normed spaces [2,3,12,18,19]. For Euclidean space starting from dimension 3 several
classesofpolytopessuchas
• polytopes in Rd with d 2 vertices, d 2 facets, or more vertices than facets [3,
+ +
Corollary7],
• centrallysymmetricpolytopes[14,Claim2],
• simplepolytopes,i.e.,polytopesinRd whereeachvertexisincidentto d edges(like
polytopalprisms,forinstance)[3,Corollary8],
• pyramidswithpolytopalbase[2,Theorem1],andinparticularsimplices[18,19]
wereprovedtobenotreduced. Thepurposeofthepresentarticleistwo-fold. Afterproving
a novel necessary condition for reduced polytopes in three-dimensional Euclidean space
inSection3,wepresentareducedpolytopeinR3inSection4. Thevalidityofourexample
canbecheckedusingthealgorithmprovidedinSection5.
2 Notation and basic results
Throughout this paper, we work in d-dimensional Euclidean space, that is, the vector
spaceRd equippedwiththe innerproduct x y : d x y and thenorm x : p x x ,
〈 | 〉 = i 1 i i k k = 〈 | 〉
=
P
2
where x (x ,...,x )and y (y ,...,y )denotetwopointsinRd. Asubset K Rd issaid
1 d 1 d
= = ⊆
tobeconvexifthelinesegment
[x,y]: {λx (1 λ)y : 0 λ 1}
= + − ≤ ≤
is contained in K for all choices of x,y K. Convex compact subsets of Rd having non-
∈
emptyinteriorarecalledconvexbodies. Thesmallestconvexsupersetof K Rd iscalled
⊆
itsconvexhullco(K),whereasthesmallestaffinesubspaceofRd containingK isdenoted
by aff(K), the affine hull of K. The affine dimension dim(K) of K is the dimension of its
affinehull. Thesupportfunction h(K, ):Rd Rof K isdefinedby
· →
h(K,u): sup{ u x : x K}.
= 〈 | 〉 ∈
For u Rd\{0},thehyperplaneH(K,u): x Rd : u x h(K,u) isasupportinghyper-
∈ = ∈ 〈 | 〉=
planeof K. Thewidthof K indirection u Rd,definedby
∈© ª
w(K,u): h(K, u) h(K,u)
= − +
equalsthedistanceof thesupportinghyperplanes H(K, u)multipliedby u . Themin-
± k k
imumwidthof K isω(K): inf{w(K,u): u 1}. Apolytopeistheconvexhulloffinitely
= k k=
many points. The boundary of a polytope consists of faces, i.e., intersections of the poly-
topewithitssupportinghyperplanes. Weshallrefertofacesofaffinedimension0,1,and
d 1asvertices,edges,andfacets,respectively. Facesofpolytopesarelower-dimensional
−
polytopes and shall be denoted by the list of their vertices. (A face which is denoted in
this way can be reconstructed by taking the convex hull of its vertices.) By definition,
attainmentoftheminimalwidthofapolytopeP isrelatedtoabinaryrelationonfacesof
P calledstrictantipodality,see[2].
Definition 2.1. Let P Rd be a polytope. Distinct faces F , F of P are said to be
1 2
⊆
strictly antipodal if there exists a direction u Rd, u 1, such that H(P,u) P F
1
∈ k k= ∩ =
and H(P, u) P F .
2
− ∩ =
GritzmannandKlee[7,(1.9)]formulatedanecessaryconditiononstrictlyantipodalpairs
whosedistanceequalstheminimumwidth. Here, F F {x y : x F ,y F } denotes
1 2 1 2
+ = + ∈ ∈
theMinkowskisumofsetsF ,F Rd. (ThesetF F isdefinedanalogously,F vshall
1 2 1 2 1
⊆ − ±
beusedasanabbreviationforF {v}wheneverv Rd,and,usingtheaboveconventions,
1
± ∈
v v v v [v ,v ] [v ,v ]for v ,...,v Rd.)
1 2 3 4 1 2 3 4 1 4
− = − ∈
Theorem 2.2. Suppose that P Rd is a polytope with non-empty interior, and that F
1
⊆
andF areastrictlyantipodalpairoffacesof P whosedistanceisequaltoω(P). Then,
2
dim(F F ) d 1, (1)
1 2
+ = −
withdim(F ) dim(F ) d 1when P iscentrallysymmetric.
1 2
= = −
Forarbitrarysubsets A,B Rd,weshalldenoteby
⊆
ρ(A,B) inf{ x y : x aff(A),y aff(B)}
= k − k ∈ ∈
3
theminimaldistancebetweenpointsofaff(A)andaff(B). InthesituationofTheorem2.2,
ρ(F ,F )isthen said distancebetween therespectiveparallelsupportinghyperplanesof
1 2
P.
ThefollowingdefinitionbyHeil[8]iscentraltothepresentinvestigation.
Definition 2.3. A convex body K is said to be reduced if we have ω(K ) ω(K) for all
′
<
convexbodiesK (K.
′
Reduced polytopes can be characterized using vertex-facet distances, see [3, Theorem 4]
and[12,Theorem1]forthefollowingresult.
Theorem 2.4. A polytope P Rd is reduced if and only if for every vertex v of P, there
⊆
exists a strictlyantipodalfacet F of P such that the distancebetween v and aff(F) equals
ω(P).
Strongly related, there is also the following necessary condition on the orthogonal pro-
jection of a vertex onto one of its strictly antipodal facets at the correct distance, see [2,
Lemma2].
Theorem 2.5. Assume that P Rd is a reduced polytope. Then for every vertex v of P
⊆
thereexistsafacetF of P suchthat{v}isstrictlyantipodalto F,theorthogonalprojection
w of v ontoaff(F)liesintherelativeinteriorof F,andthedistancefrom v to wisequalto
ω(P).
3 A class of non-reduced polytopes
Inthissection,weprovethefollowingnecessaryconditiononreducedpolytopesinthree-
dimensionalEuclideanspace.
Theorem 3.1. Suppose that P R3 is a reduced polytope. Let F be a facet of P with
⊆
edges a a ,...,a a ,a a , and let v be a vertex of P. Suppose that in this clockwise
1 2 k 1 k k 1
−
order, vv ,...,vv , vv ,...,vv ,..., vv ,...,vv , where k,i ,...,i denote positive
1,1 1,i1 2,1 2,i2 k,1 k,ik 1 k
integers, are the edges incident to v. For any j {1,...,k} and l 1,...,i 1 , let F be
j j,l
∈ ∈ −
the facet incident to vv and vv . For j {1,...,k 1}, let F be the facet incident to
j,l j,l+1 ∈ − j,ij© ª
vv and vv . Finally, denote by F be the facet incident to vv and vv . Then
j,ij j+1,1 k,ik k,ik 1,1
thefollowingconditionscannotbetrueatthesametime:
(a) ThefacetF andthevertexv arestrictlyantipodal,andρ(v,F) ω(P).
=
(b) Forany j {1,...,k},theedgesvv anda a arestrictlyantipodal. (Takea a .)
j,1 j 1 j 0 k
∈ − =
(c) For any j {1,...,k}, the facets F ,...,F are strictly antipodal to a . Moreover,
∈ j,1 j,ij j
thereisanumber l 1,...,ij suchthatρ(aj,Fj,l) ω(P).
∈ =
© ª
We preparethe proof of Theorem 3.1 by three lemmas which rely on the geometry of the
counterpartsofconvexpolygonsinsphericalgeometry. Inordertoavoidambiguity,wefix
4
v
v k,ik
k,1
a
3
Fk,ik v1,1
a v
2 F
F 1,i1
v
2,1
a1 F2,i2 v2,i2 F2,1
a
k
Figure1.NotationofTheorem3.1: facesofP.
therequirednotionsandnotation. Theballandthespherewithcenter x R3 andradius
∈
α 0shallbedenotedby
>
B(x,α): y R3 : y x α ,
= ∈ k − k≤
S(x,α): ©y R3 : y x αª,
= ∈ k − k=
respectively. The open half-space H ©: y R3 : u y ªα and its closed counterpart
u<,α = ∈ 〈 | 〉<
H : y R3 : u y α are bounded by the hyperplane H : y R3 : u y α ,
u≤,α = ∈ 〈 | 〉≤ © ª u=,α = ∈ 〈 | 〉=
where u S(0,1) denotes the outer normal unit vector of these three sets. Now fix a
©∈ ª © ª
sphere S S(x ,α). Agreatcircleof S istheintersectionof S andahyperplane H .
Ahemisp=hereo0f S istheintersectionofS andanopenhalf-spaceH . Let x,yu=,〈uS|xb0〉e
containedinahemisphereofS. Thereisexactlyonegreatcircle,denu<o,〈tue|dx0b〉yD ,pa∈ssing
x,y
through xand y,anditisdividedintotwoconnectedcomponentsby xand y,oneofwhich
lies in the same hemispherelike x and y. This connected component is called the arc xy
whoselengthshallbedenotedby xy.
| |
A cap C of S is the intersection of S and a closed half-space H with α β 0. ThØe
u≤,β − < <
(spherical) boundary sbd(C) of a cap C S H is the circle S H . The center of
= ∩ u≤,β ∩ u=,β
the cap C S H is the singleton S H . A subset A of S which is contained in
= ∩ u≤,β ∩ u=,−α
a hemisphere is said to be spherically convex if for every choice of x,y A, the arc xy is
∈
fully contained in A. Equivalently, a subset A of S is spherically convex if and only if
the positive hull pos(A x ): {λu : λ 0,u A x } is a convex set. A sphericalpolØygon
0 0
− = ≥ ∈ −
is then the smallest spherically convex set containing a given finite subset of S. An arc
xy S touchesacapC ifaff{x ,x,y} C isasingletonandiscontainedin xy.
0
⊆ ∩
Lemma 3.2. Let S be a hemisphere of S, C S be a cap, x S \C. Furthermore, let
Ø ′ ⊆ ′ ∈ ′ Ø
y ,y sbd(C)suchthatthearcs xy and xy touchC. Then xy xy .
1 2 1 2 1 2
∈ | |=| |
TheproofofLemma3.2islefttotheinterestedreader.
Ù Ù
Lemma 3.3. Let S beahemisphereof S S(x ,α), C S beacap,a, b S \C. Assume
′ 0 ′ ′
= ⊆ ∈
thatab C containsatleasttwopoints. Furthermore,let x,y sbd(C)besuchthataxand
∩ ∈
bytouchC. Then ab ax by.
| |≥| |+| |
Ø Ø
Ø
5
y
1
x
C y
2
Figure2.IllustrationofLemma3.2.
t
b a
′ ′
x y
z
C ′
′
t b
′
a
z
C
Figure3.IllustrationofLemma3.3.
Proof. Without loss of generality we assume that x and y lie in the same hemisphere
boundedby D . Choose t D D suchthatthat x atand y bt,i.e.,thepoints x,
a,b a,x b,y
∈ ∩ ∈ ∈
y, tlieonthesamehemisphereboundedbyD . ConsidertheincircleC ofthespherical
a,b ′
trianglewithverticesa,b,andt,thatis,thelargestcapconØtainedintØhistriangle. Denote
its center by z. The incircle C touches all sides of the spherical triangle; we thus set
′ ′
a : C bt, b : C at, and t : C ab. Note that b xt. (Else, if z denotes the
′ ′ ′ ′ ′ ′ ′
= ∩ = ∩ = ∩ ∈
center of the cap C, the arcs z b and xz intersect in a point u. Hence ux zx απ/2.
©Buªt ux andØub©′ aªre bothØorth′og′o©naªl to Da,tØ, i.e., ux ub′ Øαπ/2,|a c|o≤n|tra|d<iction.)
| | = =
Analogously,wehavethat a Ùyt. TherØefore,
′∈ ¯ ¯
Ø Ù ¯ ¯
ax bØy ab′ ba′ at′ bt′ ab .
| |+| |≤ + = + =| |
¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯
NotethatwehaveusedLemma3.2¯int¯he¯firs¯teq¯ual¯ity¯. ¯
Lemma3.4. Let p ,...,p ,k 4,be(inthiscyclicorder)theverticesofasphericalpolygon
1 k
≥
lyinginahemisphereS ,andletC S beacapoverlappingwiththeinteriorofP. Assume
′ ′
⊆
that p p touchesC,andthat p p and p p aredisjointtoCortouchC. Furthermore,
k 1 1 2 k 1 k
−
assumethatforsome i {2,...,k 2}, p p hasnon-emptyintersectionwithC. Then
i i 1
∈ − +
Û Û Ü
k 2
p p − Üp p p p p p
k 1 j j 1 1 2 k 1 k
| |+j 2 + ≥| |+| − |
X= ¯ ¯
¯ ¯
6
pi q2 qk−1 rk 1 pi+1
−
r
2
qk pk−1
q
p 1
2
C
p
k
p1 q0
Figure4.IllustrationofLemma3.4.
Proof. The arc p p has non-empty intersection with C. (Otherwise, C is a subset of
2 k 1
−
the spherical polygon with vertices p , p , p , and p , i.e., C and p p are disjoint.)
1 2 k 1 k i i 1
− +
DuetothetriangÜleinequality,
Ü
k 2
−
p p p p .
j j 1 2 k 1
j 2 + ≥| − |
X= ¯ ¯
¯ ¯
Thus,itremainstoshowthat
p p p p p p p p .
1 k 2 k 1 1 2 k 1 k
| |+| − |≥| |+| − |
Let q sbd(C) be such that the arc p q touches C, i {1,2,k 1,k}, q ,q p p . Let
i i i 1 k 1 k
∈ ∈ − ∉
q sbd(C) be such that the arc p q does not intersect p q . Analogously, let q
2 2 2 1 1 k 1
∈ − ∈
sbd(C) be such that the arc p q Údoes not intersect p q . Let q be the inÛtersection
k 1 k 1 k k 0
− −
pointofsbd(C)and p p . ThegreaÚtcirclethrough p andqÚintersects p q inapointr .
1 k 1 1 2 2 2
Similarly, the great circle throÜugh p and q intersectsÛp q in a point r . Using
k k k 1 k 1 k 1
− − −
thetriangleinequalÛity,Lemma3.2,andLemma3.3,weobtain Ú
Ü
p p p p p q q r r p p q q r r p
1 2 k k 1 1 1 1 2 2 2 k k k k 1 k 1 k 1
| |+| − |≤| |+| |+| |+| |+| − |+| − − |
p q q r r p p q q r r p
1 0 2 2 2 2 k 0 k 1 k 1 k 1 k 1
=| |+| |+| |+| |+| − − |+| − − |
p p q p q p
1 k 2 2 k 1 k 1
=| |+| |+| − − |
p p p p .
1 k 2 k 1
≤| |+| − |
Thiscompletestheproof.
Proof of Theorem 3.1. Suppose that P is a reduced polytope which satisfies the condi-
tions(a),(b),and(c)ofTheorem3.1. Clearly,thefacetsofthedifferencepolytopeP P P
′
= −
havetheform F F with F and F beingstrictlyantipodalfacesof P satisfyingEqua-
1 2 1 2
−
tion (1). Among the facets of the difference polytope P P P, there are F F v,
′ 0
= − = −
G a F , P a a vv . Notice that F and G are congruent to F and F ,
j,l j j,l j j 1 j j,1 0 j,l j,l
= − = − −
7
G
2,i2
d
2
c
2,i2
G
2,1
c
2,1 b
3
P2 b2 F0
d
1 b
1
G1,i1 bk
c P
1,1 1 P
k
d
k
Figure5.NotationofTheorem3.1: facesofP .
′
respectively, and that P is a parallelogram. For j {1,...,k} and l 1,...,i , denote by
j j
∈ ∈
b a v theverticesof F , andlet c a v , d a v . Obviously, F and P
j= j− 0 j,l= j− j,l j= j− j+1,1 © ª 0 j
shareacommonedge b b a a v.
j 1 j j 1 j
− = − −
Wehave
ρ(0,F ) ω(P),
0
=
ρ(0,Pj) ω(P) for j {1,...,k},
≥ ∈
and ρ(0,Gj,l) ω(P) for j {1,...,k} and l 1,...,ij .
≥ ∈ ∈
© ª
Moreover, for each j {1,...,k}, thereis l 1,...,ij such that ρ(0,Gj,l) ω(P). Next, we
∈ ∈ =
provetheinequalities
© ª
∡b b b ij−1∡c b c ∡c b d ∡b b c ∡d b b (2)
j−1 j j+1+ j,l j j,l+1+ j,ij j j≥ j−1 j j,1+ j j j+1
l 1
X=
for j {1,...,k}, where we use the abbreviation ∡xyz arccos 〈x−y|z−y〉 for the angle
∈ = x y z y
k − kk − k
between vectors x y and z y. Denote by q the orthogonal pr³ojection of´0 onto aff(F0)
− −
and consider the ball B B(0,ω(P)) with center 0 and radius ω(P) and the spheres Sj
= =
S b , b q , j {1,...,k}, withcenter b andradiusequal tothelength of thesegment
j j j
k − k ∈
b q. Let C S B and Q S (b pos(P b )). Then Q is a spherical polygon
¡j j =¢ j∩ j = j∩ j+ ′− j j
containedinahemisphereof S . (Choose asupportinghyperplane H of P with H P
j ′ ′
∩ =
b . Then H S isthesphericalboundaryofanadmissiblehemisphere.) Since
j j
∩
© ª
C S B B P ,
j j ′
= ∩ ⊆ ⊆
wealsohave
b pos(C b ) b pos(P b ),
j j j j ′ j
+ − ⊆ + −
andthus
C S (b pos(C b )) S (b pos(P b )) Q .
j j j j j j j ′ j j
= ∩ + − ⊆ ∩ + − =
8
Denoteby p theintersectionpointofS andb pos(b b ),by p , l 2,...,i 1 ,
j,1 j j j 1 j j,l j
+ − − ∈ +
theintersectionpointof S and b pos(c b ),by p theintersectionpointof S
j j+ j,l−1− j j,ij+2 © ªj
and b pos(d b ), and by p the intersection point of S and b pos(b b ).
j+ j− j j,ij+3 j j+ j+1− j
Since ρ(0,Gj,l 1) ω(P) for some l 2,...,ij 1 , the arc pj,lpj,l 1 touches the cap
− = ∈ + +
C . Analogously, p p touches C . Notice that p p and p p inter-
j j,ij+3 j,1 ©j ª j,1 j,2 j,ij+2 j,ij+3
sect Cj in at most one point each because ρ(0,Pj) ω(P) and ρÜ(0,Pj 1) ω(P). There-
≥ + ≥
fore, the assumptionÜs from Lemma 3.4 hold. In particuÜlar, since the lenÜgths of arcs in
Lemma 3.4 correspond to angles in Equation (2), the latter equation is true. Note that
∡d b b ∡b b c π for all j {1,...,k} because P is a parallelogram. Adding
j j j 1 j j 1 i 1,1 j
+ + + + = ∈
theinequalities(2)for j {1,...,k}yields
∈
k ∡b b b ij−1∡c b c ∡c b d k ∡b b c ∡d b b kπ.
j 1Ã j−1 j j+1+ l 1 j,l j j,l+1+ j,ij j j!≥j 1 j−1 j j,1+ j j j+1 =
X= X= X= ¡ ¢
Hence,
k ∡b b b ij−1∡c b c ∡c b d ∡b b c ∡d b b 2kπ (3)
j 1Ã j−1 j j+1+ l 1 j,l j j,l+1+ j,ij j j+ j−1 j j,1+ j j j+1!≥
X= X=
This last inequalitycontradictsthe fact that the angles occuringon the left-hand side of
Equation(3)areinternalanglesofthefacetsadjacentto b atthisvertex.
j
Note that the idea of the proof of Theorem 3.1 is similar to Steinitz’s approach in [23],
whereheconstructedanexampleofnon-circumscribablepolytopeinR3.
SpecifyingTheorem3.1to i ... i 1,weobtainthefollowingresult.
1 k
= = =
Corollary3.5. SupposethatP R3 isareducedpolytope. LetF beafacetofP withedges
⊆
a a ,...,a a ,a a , and let v be a vertex of P. Suppose that in this clockwise order,
1 2 k 1 k k 1
−
vv ,...,vv ,..., vv arethe edges incidentto v. For any j {1,...,k 1}, let F be the facet
1 2 k j
∈ −
incidentto vv andvv . Finally,denoteby F bethefacetincidentto vv and vv . Then
j j 1 k k 1
+
thefollowingconditionscannotbetrueatthesametime:
(a) ThefacetF andthevertexv arestrictlyantipodal,andρ(v,F) ω(P).
=
(b) Forany j {1,...,k},theedgesvv anda a arestrictlyantipodal. (Take a a .)
j j 1 j 0 k
∈ − =
(c) Forany j {1,...,k},thefacetFj isstrictlyantipodaltoaj andρ(aj,Fj) ω(P).
∈ =
Now assume that P R3 is a combinatorially self-dual polytope, i.e., there exists an
⊆
inclusion-reversing bijective map φ from the face lattice of P onto itself, and for each
face F of P, φ(F) is the unique antipodal face of F. Then the conditions (a), (b), and (c)
inCorollary3.5aresatisfiedateachvertexofP whichrendersthepolytopenon-reduced.
In particular, this applies to the case of pyramids in three-dimensional Euclidean space
whichbelongtoseveralclassesofnon-reducedpolytopesmentionedinSection1.
9
3 6 5
4 4
3
6 5 8 7
8 11 2 1 9
12 11 7 10 12 10 12 10
9
4
2 1
2
1 8 7
Figure6.AreducedpolytopeP withvertexnumbers: obliqueview(left),topview(middle),front
view(right)
4 A reduced polytope
In contrast to the various classes of polytopes which are shown to be non-reduced in the
literatureandinSection3,wepresentareducedpolytopeP now. Considerthepoints
v : (r,0, t), v : ( r,0, t), v : (0,r,t), v : (0, r,t),
1 2 3 4
= − = − − = = −
v : (h,x,s), v : ( h,x,s), v : (h, x,s), v : ( h, x,s),
5 6 7 8
= = − = − = − −
v : (x,h, s), v : (x, h, s), v : ( x,h, s), v : ( x, h, s).
9 10 11 12
= − = − − = − − = − − −
For properlychosen parameters t,x,s,h,r 0the points v ,...,v are thevertices of our
1 12
>
polytope P. The combinatorial structureof our polytope is shown in Figure 6. The poly-
tope P possesses the same symmetry as the Johnson solid J (however, not the same
84
combinatorialstructure). Hence, it issufficienttocontrolfew facet-vertex andedge-edge
distances. Infact,wearegoingtosolvetheequations
ρ(v , v v v v ) 1, ρ(v v , v v ) δ , ρ(v v , v v ) δ ,
1 3 11 12 4 1 2 3 4 1 1 9 4 8 3
= = =
ρ(v , v v v ) 1, ρ(v v , v v ) δ ,
5 2 8 12 1 5 4 8 2
= =
with respect to t,x,s,h,r. Here, δ ,δ ,δ 1 are suitably chosen. By introducing the
1 2 3
≥
normalvectors
n : (v v ) (v v ), n : (v v ) (v v ),
1 11 3 12 3 4 1 5 4 8
= − × − = − × −
n : (v v ) (v v ), n : (v v ) (v v ),
2 8 2 12 2 5 1 9 4 8
= − × − = − × −
n : (v v ) (v v ) (0,0,4r2),
3 1 2 3 4
= − × − =
where u w denotestheusualcrossproductofthevectors u,w R3,theseequationscan
× ∈
berewrittenas
n v v 2 n 2 0, n v v 2 δ2 n 2 0, n v v 2 δ2 n 2 0,
〈 1| 1− 3〉 −k 1k = 〈 3| 3− 1〉 − 1k 3k = 〈 5| 1− 4〉 − 3k 5k =
n v v 2 n 2 0, n v v 2 δ2 n 2 0.
〈 2| 5− 2〉 −k 2k = 〈 4| 1− 4〉 − 2k 4k =
10