Table Of ContentTHE MEAN WIDTH OF CIRCUMSCRIBED RANDOM
POLYTOPES
9
0 KA´ROLYJ.BO¨RO¨CZKYANDROLFSCHNEIDER
0
2
n
Dedicated to Professor Tibor Bisztriczky on the occasion of his 60th birthday
a
J
1 Abstract. For a given convex body K in Rd, a random polytope K(n) is
2
defined (essentially) as the intersection of n independent closed halfspaces
containingK andhavinganisotropicand(inaspecifiedsense)uniformdistri-
]
G bution. Weproveupperandlowerbounds,ofoptimalorders,forthedifference
ofthemeanwidthsofK(n)andK,asntendstoinfinity. Forasimplicialpoly-
M
topeP,apreciseasymptoticformulaforthedifferenceofthemeanwidthsof
P(n) andP isobtained.
.
h
t
a 1. Introduction and Results
m
Theconvexhullofnindependent,uniformlydistributedrandompointsinagiven
[
convexbodyK ind-dimensionalEuclideanspaceisatypeofrandompolytopethat
1 hasbeenstudiedextensively(basicreferencesarefoundinthesurveys[23]and[22],
v see also [13]). As in the seminal papers of R´enyi and Sulanke [17, 18] (restricted
3
to the planar case), which initiated this line of research,most of the investigations
4
deal with asymptotic results, for n tending to infinity. In a third paper, R´enyi
3
3 and Sulanke [19] have studied a dual way of generating random polytopes related
. to a convex body K (again in the plane), by taking intersections of independent
1
random closed halfspaces containing the body. Subsequently, this approach has
0
9 attractedlessattentionthantheconvexhullsofrandompoints,althoughitdeserves
0 similarinterest. Inthe presentpaper,we obtainsomeresults onrandompolytopes
: generated in the second way.
v
i Throughout the following, K is a convex body with interior points in d-dimen-
X sional Euclidean space Rd (d ≥ 2). For any notions on convexity in this paper,
r see the monographs Schneider [21] or Gruber [14]. Let Bd be the unit ball of
a
Rd with center at the origin; then K := K +Bd is the parallel body of K at
1
distance 1. By H we denote the space (with its usual topology) of hyperplanes
in Rd, and H is the subspace of hyperplanes meeting K but not the interior of
K 1
K. For H ∈ H , the closed halfspace bounded by H that contains K is denoted
K
by H−. The measure µ is the motion invariant Borel measure on H, normalized
so that µ({H ∈ H : H ∩ M 6= ∅}) is the mean width W(M) of M, for every
convex body M ⊂ Rd. Let 2µ be the restriction of µ to H . Since µ(H ) =
K K K
W(K +Bd)−W(K) = W(Bd) = 2, the measure µ is a probability measure.
K
2000 Mathematics Subject Classification. Primary52A22,Secondary60D05, 52A27.
Key words and phrases. Randompolytope, meanwidth,approximation.
The first author was supported by OTKA grants 068398 and 049301 and by the EU Marie
CurieprojectBudAlgGeo(MTKD-CT-2004-002988). BothauthorsweresupportedbyEUMarie
CurieprojectsDiscConvGeo(MTKD-CT-2005-014333)andPHD(MCRN-2004-511953).
1
2 KA´ROLYJ.BO¨RO¨CZKYANDROLFSCHNEIDER
For n ∈ N, let H ,...,H be independent random hyperplanes in Rd (H-valued
1 n
random variables on some probability space (Ω,A,P)), each with distribution µ .
K
Theintersection n H− isarandompolyhedralset,possiblyunbounded. Weput
i=1 i
n
T
K(n) := H−∩K
i 1
i=1
\
and ask for EW(K(n)), where E denotes mathematical expectation. Alternatively,
we might consider E W(K(n)), the conditional expectation of W(K(n)) under the
1
condition that n H− ⊂ K . Since EW(K(n)) = E W(K(n)) + O(γn) with
i=1 i 1 1
γ ∈ (0,1), as is easy to see, there is no difference in the asymptotic behaviors of
T
both quantities, as n → ∞. We also remark that, for the asymptotic results, the
parallel body K could be replaced by any other convex body containing K in its
1
interior; this would only affect some normalization constants.
The preceding model has to be distinguished from the one where n indepen-
dent random points are chosen from the boundary of K, and the intersection of
the supporting halfspaces of K at these points is the random polyhedron under
consideration. For this model and sufficiently smooth convex bodies, B¨or¨oczky
andReitzner[8]havederivedasymptoticexpansionsoftheexpectationsofvolume,
surface area and mean width.
For comparison, we mention first some results involving convex hulls of random
points. LetK be the convexhullof n independent, uniformly distributed random
n
points in the convex body K. Throughout this paper, c ,c ,... are positive con-
1 2
stants that depend only on K and d. In writing n ≥ n , we indicate that a result
0
is true for all sufficiently large n∈N. There exist c ,...,c such that, for n≥n ,
1 4 0
(1) c n−2/(d+1) <W(K)−EW(K )<c n−1/d,
1 n 2
(2) c n−1lnd−1n<V(K)−EV(K )<c n−2/(d+1),
3 n 4
where V denotes the volume. The inequalities (1) are due to Schneider [20], and
(2) to B´ar´any and Larman [7]. The orders are best possible, being attained in
(1)(left)and(2)(right)bysufficientlysmoothbodies,andin(1)(right)and(2)(left)
by polytopes.
If one sets about obtaining analogous results for random polytopes obtained
as intersections of halfspaces, the idea of dualizing immediately comes to mind.
Supposing that o∈intK, polarization with respect to the unit sphere sends K to
its polar body K∗, and it interchanges hyperplanes not meeting intK with points
in K∗ (hence, mean width and volume should interchange their roles); cum grano
salis, intersections of halfspaces correspond to convex hulls of points. Measures
on hyperplanes correspond to measures on points; however, uniform measures do
not correspond to uniform measures, and mean width and volume are not exactly
related under polarity. Nevertheless, this approach can be put to work in some
cases,orargumentsmayfindtheirdualanalogsinamoreheuristicway. Inthismore
orlessvaguesense,dualityhasbeenappliedtopolygonsandvertexnumbersinthe
planebyZiezold[25],andtosmoothandgeneralconvexbodiesinhigherdimensions
byKaltenbach[15]. In particular,Kaltenbachhasestablisheda counterpartto (1),
namely
(3) c n−2/(d+1) <EV(K(n))−V(K)<c n−1/d.
5 6
We obtain here a counterpart to (2), again with optimal orders.
MEAN WIDTH OF RANDOM POLYTOPES 3
Theorem 1.1. For n≥n ,
0
(4) c n−1lnd−1n<EW(K(n))−W(K)<c n−2/(d+1).
7 8
The right side follows from a result of independent interest.
Theorem 1.2. For each n∈N, the functional K 7→EW(K(n))/W(K) attains its
maximum at balls.
The following precise asymptotic formula is a counterpart to a result of Affen-
tranger and Wieacker [2].
Theorem 1.3. If P is a simplicial polytope in Rd with r facets, then, as n→∞,
2rd lnd−1n
(5) EW(P(n))−W(P)∼ .
(d+1)d−1 n
For the polytope P in Theorem 1.3, we also obtain asymptotic results for the
numbers of vertices and facets. We denote by f (P(n)) the number of k-faces of
k
P(n) thatarecontainedintheinteriorofP (recallthatP(Pn) 6⊂P )=O(γn)with
1 1
0<γ <1). Then
rdd
(6) Ef (P(n))∼ M (∆ )lnd−1n,
0 1 d−1
d!
where the constant M (∆ ) is given by (15), and
1 d−1
rd
(7) Ef (P(n))∼ lnd−1n.
d−1 (d+1)d−1
Remark. Based on the paper of B´ar´any and Buchta [6] generalizing the results
of Affentranger and Wieacker [2], one can most probably extend Theorem 1.3 as
follows. Let P be any polytope in Rd. We write T(P) to denote the number of
flags (or towers) of P; namely, the number of chains F ⊂ ... ⊂ F where F is
0 d−1 i
an i-face of P. Then, as n→∞,
2T(P) lnd−1n
EW(P(n))−W(P) ∼ ,
(d+1)d−1(d−1)! n
T(P)
Ef P(n) ∼ lnd−1n,
d−1 (d+1)d−1(d−1)!
T(P)dd
Ef P(n) ∼ M (∆ )lnd−1n.
0 (d!)2 1 d−1
Moreover, it is well known that M (∆ ) = 1 and M (∆ ) = 1 , and Buchta and
1 1 3 1 2 12
Reitzner [9] proved M (∆ )= 13 − π2 .
1 3 720 15015
2. Proofs of Theorems 1.1, 1.2
First,wefixsomemorenotation. Inthefollowing,Sd−1 :={x∈Rd :hx,xi=1}
(where h·,·i denotes the scalar product) is the unit sphere of Rd, λ is Lebesgue
measure on Rd, and σ is the spherical Lebesgue measure on Sd−1.
For sets A ,...,A and points x ,...,x in Rd, m,k ∈N , we write
1 m 1 k 0
[A ,...,A ,x ,...,x ]:=conv(A ∪...∪A ∪{x ,...,x }).
1 m 1 k 1 m 1 k
Before the proofs, we want to substantiate the remark, made in the introduc-
tion, about the comparison between EW(K(n)) and the conditional expectation
4 KA´ROLYJ.BO¨RO¨CZKYANDROLFSCHNEIDER
E W(K(n)). Clearly, there are finitely many hyperplanes E ∈ H , j = 1,...,k,
1 j K
such that k E− ⊂ K + (1/2)Bd. We can choose neighborhoods N of E ,
j=1 j j j
j = 1,...,k, of equal measure µ (N ) =: α ∈ (0,1), such that any hyperplanes
K j
T
H ,...,H with H ∈ N , j = 1,...,k, satisfy k H− ⊂ K +Bd. Now let
1 k j j j=1 j
H ,...,H be independent random hyperplanes with distribution µ , and define
1 n K
n H− := P . The event P 6⊂ K occurs onTly if one of the events A :=
i=1 i n n 1 j
{H ∈/ N for i = 1,...,k} occurs. It follows that P(P 6⊂ K ) ≤ P( k A ) ≤
T i j n 1 j=1 j
k P(A )=k(1−α)n. From
j=1 j S
P EW(K(n))−(1−P(P 6⊂K ))E W(K(n))
n 1 1
= W(K(n))dP≤W(K )P(P 6⊂K )
1 n 1
ZPn6⊂K1
we now conclude that EW(K(n))−E W(K(n))=O(γn) with γ ∈(0,1).
1
Turning to the proofofTheorem1.1, we firstrecallthat B´ara´nyandLarman[7]
proved (2) after establishing the following general result. For x ∈ K, let v(x) be
the minimal volume that a closed halfspace with x in its boundary cuts off from
K. For (small) t>0, let
(8) K(t):={x∈K :v(x)≤t}.
In [7], the existence of positive constants c ,c with
9 10
(9) c V(K(1/n))<V(K)−EV(K )<c V(K(1/n))
9 n 10
for n≥n was proved. Part of this approachwill now be ‘dualized’.
0
For x ∈ Rd, let K := [K,x], and let w(x) be the µ-measure of the set of
x
hyperplanes separating K and x, thus w(x) = W(K )−W(K). For a hyperplane
x
H ∈H , let
K
m(H):=min{w(x):x∈H},
and for t>0 (sufficiently small), define
H (t):={H ∈H :m(H)≤t}.
K K
It is convenient to describe this set of hyperplanes in a different way. For this, put
K[t]:={x∈Rd :w(x)≤t}.
Letx,y ∈K[t],λ∈[0,1],andz ∈K . Thenz =(1−α)[(1−λ)x+λy]+αk
(1−λ)x+λy
with suitable k ∈K and α∈[0,1]. It follows that
z =(1−λ)[(1−α)x+αk]+λ[(1−α)y+αk]∈(1−λ)K +λK ,
x y
thus K ⊂(1−λ)K +λK . This gives
(1−λ)x+λy x y
W(K )≤W((1−λ)K +λK )=(1−λ)W(K )+λW(K )
(1−λ)x+λy x y x y
≤(1−λ)(W(K)+t)+λ(W(K)+t)=W(K)+t,
hence w((1−λ)x+λy)≤t and thus (1−λ)x+λy ∈K[t]. This shows that K[t] is
convex.
Now let H ∈ H . If H ∩K[t] 6= ∅, then H contains a point x with w(x) ≤ t,
K
hence m(H) ≤ t and, therefore, H ∈ H (t). If H ∩K[t] = ∅, then every x ∈ H
K
satisfies w(x) > t, and since m(H) is an attained minimum, also m(H) > t and
MEAN WIDTH OF RANDOM POLYTOPES 5
hence H ∈/ H (t). Thus, H (t) is the set of hyperplanesmeeting the convexbody
K K
K[t] but not the interior of K. In particular,
µ(H (t))=W(K[t])−W(K).
K
The left inequality in (9) admits a straightforwarddualization, as alreadynoted
by Kaltenbach [15]. The following argument, which we give for the reader’s con-
venience, is the exact dual analog of that in [7, p. 283]. Let H ,...,H and
1 n
K(n) be as in the introduction, n > d. Let H ∈ H and choose x ∈ H such
K 0
that w(x ) = m(H). If H does not separate x and K, for i = 1,...,n, then
0 i 0
H ∩K(n) 6=∅, hence
P(H ∩K(n) 6=∅)≥(1−m(H))n.
For small t>0, we obtain
EW(K(n))−W(K)= 1{H∩K(n) 6=∅}µ(dH)dP
Z ZHK
= P(H ∩K(n) 6=∅)µ(dH)≥ (1−m(H))nµ(dH)
ZHK ZHK
> 1{m(H)≤t}(1−t)nµ(dH)
ZHK
=(1−t)nµ(H (t))=(1−t)n(W(K[t])−W(K)).
K
The choice t=1/n gives
(10) c (W(K[1/n])−W(K))<EW(K(n))−W(K).
11
Next, we carry over results from [7] by applying them to the polar body. We
assume that o ∈ intK and let K∗ denote the polar body of K. We write the
points of Rd in the form ru with u ∈ Sd−1 and r ≥ 0 and the hyperplanes of Rd
in the form H(u,t) := {x ∈ Rd : hx,ui = t} with u ∈ Sd−1 and t ≥ 0. The map
ϕ:Rd\{o}→H is defined by
(11) ϕ(ru):=H(u,r−1).
Let ν denote the image measure of λ under ϕ, then
∞
ν(A)= 1{H(u,t)∈A}t−(d+1)dtσ(du)
ZSd−1Z0
for Borel sets A⊂H. For comparison, the invariant measure µ is given by
2 ∞
(12) µ(A)= 1{H(u,t)∈A}dtσ(du).
σ(Sd−1)ZSd−1Z0
Consequently, there exist positive constants c ,c such that
12 13
c ν(A)≤µ(A)≤c ν(A) if A⊂H .
12 13 K
In the following, we assume that 0 ≤ t < t , where t is chosen such that
0 0
K∗∩K∗(t )= ∅; here K∗ := (K )∗, and K∗(t ) is defined by (8). Let x∈K∗(t).
1 0 1 1 0
There is a hyperplane E through x such that λ(K∗ ∩E+) ≤ t, where E+ is the
halfspace bounded by E that does not contain o. Let H :=ϕ(x) and y :=ϕ−1(E),
then y ∈ H. The mapping ϕ maps the cap K∗ ∩E+ bijectively onto the set of
hyperplanes separating y and K, which is denoted by Hy and is a subset of H ,
K K
by the choice of t . We conclude that
0
m(H)≤µ(Hy )≤c ν(Hy )=c λ(K∗∩E+)≤c t
K 13 K 13 13
6 KA´ROLYJ.BO¨RO¨CZKYANDROLFSCHNEIDER
and hence that H ∈ H (c t). Since x ∈ K∗(t) was arbitrary, this shows that
K 13
ϕ(K∗(t))⊂H (c t); therefore,
K 13
λ(K∗(t))=ν(ϕ(K∗(t))≤ν(H (c t))≤c−1µ(H (c t)).
K 13 12 K 13
Now (10) together with this inequality gives, for n≥n ,
0
EW(K(n))−W(K)≥c µ(H (1/n))≥c c λ(K∗(1/c n)).
11 K 11 12 13
By [7, Th. 2],
λ(K∗(ǫ))≥c ǫlnd−1(1/ǫ)
14
for ǫ>0. This yields the left inequality of (4).
The right inequality of (9) relies heavily on the technique of Macbeath regions
(see B´ar´any [4, 5] for expositions of this technique and its applications), which
does not dualize in an obvious way. The proof of the right inequality of (4) can,
however, be deduced from Theorem 1.2. The latter is a counterpart to Groemer’s
[11] inequality, which says that EV(K )/V(K) is minimal if K is an ellipsoid. In
n
the subsequent proof of Theorem 1.2, ‘dualization’ becomes a bit vague: Steiner
symmetrization,whichis atoolinGroemer’sproof,is replacedbyMinkowskisym-
metrization.
Let h(K,·) be the support function of K. By (12) and the definition of the
measure µ , we have
K
1 1
µ (A)= 1{H(u,h(K,u)+t)∈A}dtσ(du)
K σ(Sd−1)ZSd−1Z0
for Borel sets A ⊂ H . We write H−(u,t) := {x ∈ Rd : hx,ui ≤ t} and use the
K
abbreviations U :=(u ,...,u ), T :=(t ,...,t ) and
1 n 1 n
P(K,U,T):=H−(u ,h(K,u )+t )∩...∩H−(u ,h(K,u )+t )∩K .
1 1 1 n n n 1
Then we get
EW(K(n))= ... W(H−∩...∩H−∩K )µ (dH )···µ (dH )
1 n 1 K 1 K n
Z Z
n
1
= W(P(K,U,T))dTσn(dU).
(cid:18)σ(Sd−1)(cid:19) Z(Sd−1)nZ[0,1]n
LetK,M ⊂Rd betwoconvexbodies. Letα∈[0,1]andx∈(1−α)P(K,U,T)+
αP(M,U,T). Then x = (1−α)y +αz with y ∈ P(K,U,T) and z ∈ P(M,U,T).
For each i∈{1,...,n}, we have hy,u i≤h(K,u )+t and hz,u i≤h(M,u )+t ,
i i i i i i
hence
hx,u i≤(1−α)(h(K,u )+t )+α(h(M,u )+t )=h((1−α)K+αM,u )+t .
i i i i i i i
Since also x∈K , we see that x∈P((1−α)K +αM,U,T). This shows that
1
(1−α)P(K,U,T)+αP(M,U,T)⊂P((1−α)K +αM,U,T)
and hence that
W(P((1−α)K +αM,U,T))≥(1−α)W(P(K,U,T))+αW(P(M,U,T)).
Inserting this in the representation of EW(K(n)), we obtain
EW([(1−α)K+αM](n))≥(1−α)EW(K(n))+αEW(M(n)).
Thus,the functionK 7→EW(K(n))isconcavewithrespecttoMinkowskiaddition,
and it is clearly invariant under rigid motions and continuous with respect to the
MEAN WIDTH OF RANDOM POLYTOPES 7
Hausdorff metric. Now the following standard argument shows that on the set of
convexbodiesofgivenmeanwidth,thefunctionEW(K(n))attainsitsmaximumat
theballs. Arotation meanofK iseveryconvexbodyoftheformK′ =m−1(δ K+
1
...+δ K) with m ∈ N and rotations δ ,...,δ of Rd. By the concavity shown
m 1 m
above and the linearity of the mean width, we have EW((K′)(n)) ≥ EW(K(n))
and W(K′) = W(K). By a theorem of Hadwiger (see [21, Theorem 3.3.2]), there
is a sequence of rotation means of K converging to a ball B. This ball satisfies
EW(B(n))≥EW(K(n)) and W(B)=W(K). We can write the result as
W(K)
(13) EW(K(n))−W(K)≤ [EW((Bd)(n))−W(Bd)],
2
which proves Theorem 1.2. The right side is of order n−2/(d+1) as n → ∞. This
can be deduced from (14) below for K = Bd, once the analogous result for the
convex hull of independent, identically distributed points in the ball Bd is known,
for the case where the Lebesgue measure (yielding the distribution of the points
and the volume functional) is replaced by Lebesgue measure with a density that
is continuous in a neighborhood of bdBd and constant on bdBd. Such a result, in
turn,isobtainedbyastraightforwardextensionofthe Lebesguemeasurecase,first
treated by Wieacker [24] and generalized by Affentranger [1]. This completes the
proof of Theorem 1.1.
3. Polarity and a useful functional
In our preparations for the proof of Theorem 1.3, we make use of the mapping
ϕ defined by (11). We assume that K is a convex body containing the origin o in
its interior. The same holds then for its polar body K∗. We define
X :=cl(K∗\K∗),
K 1
thus ϕ(X ) = H . Writing µ∗,µ∗ for the image measures of µ,µ , respectively,
K K K K
under ϕ−1, we have
1
µ∗ (A)= kxk−(d+1)dx
K σ(Sd−1)
ZA
forany BorelsetA⊂X . For a convexbody M containingK it is clear(and well
K
known, see Glasauer and Gruber [12]) that
2
W(M)−W(K)= kxk−(d+1)dx.
σ(Sd−1)ZK∗\M∗
By K∗ we denote the convex hull of K∗ and n independent random points in
n 1
X with distribution µ∗ . Thus, K∗ is stochastically equivalent to the polar body
K K n
of K(n), and we have
2
(14) EW(K(n))−W(K)= E kxk−(d+1)dx=2Eµ∗ (K∗\K∗).
σ(Sd−1) ZK∗\Kn∗ K n
For H ,...,H ∈H , we say that a (d−1)-dimensional convex compact set F
1 n K
is a proper facet ofK(n) = n H−∩K , if F =H ∩K(n) for some H which is a
i=1 i 1 i i
supporting hyperplane of K(n) intersecting intK . Further, v is a proper vertex of
1
T
K(n) ifv ∈intK and{v}istheintersectionoftheproperfacetsofK(n) containing
1
v. We write f (K(n))andf (K(n)) for the number ofproperverticesand facets,
0 d−1
respectively, of K(n).
8 KA´ROLYJ.BO¨RO¨CZKYANDROLFSCHNEIDER
Next, let K∗ = [K∗,x ,...,x ] for x ,...,x ∈ X . We say that a (d−1)-
n 1 1 n 1 n K
dimensional convex compact set F is a proper facet of K∗ if K∗∩affF =∅ and F
n 1
is the intersectionof K∗ and a supporting hyperplane of K∗. Further,some x is a
n n i
proper vertex of K∗ if x ∈/ K∗ and {x } is the intersectionof K∗ and a supporting
n i 1 i n
hyperplaneofK∗. We writef (K∗) andf (K∗) to denote the number ofproper
n 0 n d−1 n
verticesandfacets,respectively,ofK∗. IfK(n) = n ϕ(x )−∩K ,thenϕdefines
n i=1 i 1
bijective correspondences between the proper vertices of K∗ and proper facets of
T n
K(n), and between the proper facets of K∗ and proper vertices of K(n).
n
As we have seen in Section 2, there exists a number γ ∈ (0,1) depending only
on K such that with probability at least 1−O(γn) we have K∗ ⊂ intK∗. In this
1 n
case, K∗ is a polytope with vertices among the n random points determining K∗.
n n
Now we assume that P a simplicial polytope (with o in its interior), then P∗ is
a simple polytope. Similarly as in Affentranger and Wieacker [2], we consider the
functionT∗(P∗)below,whereq ≥0andP∗ =[P∗,x ,...,x ]forx ,...,x ∈X .
q n n 1 1 n 1 n P
If F is a (d − 1)-dimensional convex set whose affine hull intersects P∗ and
avoids P∗ then let v be a vertex of P∗ which is separated from P∗ by affF, and
1 F 1
where there exists a supporting hyperplane to P∗ parallel to affF. Further, let
S =[F,v ]. We write F(P∗) to denote the family of proper facets of P∗, and we
F F n n
define
T∗(P∗):= µ∗(S )q.
q n P F
F∈XF(Pn∗)
The functionals T∗(P∗) are closely related to our problem because f (P∗) =
q n d−1 n
T (P∗), and we will prove Eµ∗(P∗\P∗) ∼ ET∗(P∗) in Section 4. Now the core
0 n P n 1 n
ofthe arguments leading to Theorem1.3 is the following lemma. For this, we need
somenotation. The (d−1)-dimensionalLebesguemeasureisdenotedbyλ . For
d−1
q ≥0 and a (d−1)-dimensional compact convex set A, let
(15) M (A)=λ−d−q(A) λ ([x ,...,x ])qλd (d(x ,...,x )).
q d−1 d−1 1 d d−1 1 d
ZAd
Let ∆ be a fixed (d −1)-dimensional simplex, then M (A) = M (∆ ) for
d−1 q q d−1
any (d−1)-dimensional simplex A, by affine invariance. For an arbitrary (d−1)-
dimensionalcompactconvexsetA,thereexistsa(d−1)-dimensionalsimplexB ⊂A
such that A is contained in a translate of −(d−1)B, therefore
(16) M (A)≤(d−1)(d−1)(d+q)M (∆ ).
q q d−1
Lemma 3.1. If q ≥ 0 is an integer and P is a simplicial polytope in Rd with r
facets, then, as n tends to infinity,
r(d+q−1)!dd−1M (∆ ) lnd−1n
ET∗(P∗)∼ q+1 d−1 .
q n (d−1)!2 nq
Proof. To prove Lemma 3.1, it is sufficient to verify for any ε>0 the existence of
n and Γ depending on ε, q and P such that, if n>n , then
0 0
1 r(d+q−1)!dd−1M (∆ ) lnd−1n Γlnd−2n
(17) ET∗(P∗)> q+1 d−1 − ,
q n (1+ε)2d+2q (d−1)!2 nq nq
r(d+q−1)!dd−1M ∆ ) lnd−1n Γlnd−2n
(18) ET∗(P∗)<(1+ε)2d+2q q+1 d−1 + .
q n (d−1)!2 nq nq
Inthe restofthis section,wewriteΓ ,Γ ,...todenoteconstantsthatmaydepend
1 2
on ε, q and P.
MEAN WIDTH OF RANDOM POLYTOPES 9
Many calculations are simpler if we do them with respect to an orthonormal
basis,thereforeweintroducesomenotation. Lete ,...,e beanorthonormalbasis
1 d
ofRd,andletΩ=[o,e ,...,e ]. Forp=0,...,d,anda(d−1)-dimensionalconvex
1 d
set F, we define
e 1 if affF intersects each open ray R e , i=1,...,d,
+ i
θ˜p = and affF separates p points out of e ,...,e from o,
F 1 d
0 otherwise.
In addition, we define SF = [o,F], and η˜F denotes the distance of affF from
o. Moreover, let C be the simplex cut off by affF from d R e if affF
F i=1 ≥0 i
intersects each open rayeR e , i = 1,...,d, and let C = Ω otherwise. For s =
+ i F P
(s ,...,s )∈Rd,weewriteH(s)forthehyperplaneH thatcontainsthepointss e
1 d + i i
for i=1,...,d. It follows that (see also [2, p. 298–299e]) e
(19) V(C )=s ···s /d!,
H(s) 1 d
(20) dµ(H(s)e)=η˜Hd+(s1)(s1···sd)−2ds.
Finally, we recall the lemma in [2, p. 296]. It says that for integers k,m ≥ 0 and
p≥2, and for c∈(0,1], we have
(s ···s )k(1−cs ···s )n−m d(s ,...s )
1 p 1 p 1 p
Z(0,1]p
k! lnp−1n lnp−2n
(21) = +O
(p−1)!ck+1 nk+1 nk+1
(cid:18) (cid:19)
as n tends to infinity, where the implied constant in O(·) depends on k,m,p,c. In
addition, if p=1, then
1 k! 1 1
(22) sk(1−cs)n−mds= +O .
ck+1 nk+1 nk+2
Z0 (cid:18) (cid:19)
We choose a number ω >0 with the following four properties:
• Any edge of P∗ is of length at least 3ω.
• If y is a vertex of P∗ and w ,...,w are the points on the d edges of
y,1 y,d
P∗ meeting aty suchthat kw −yk=ω, then Ω :=[y,w ,...,w ] is
y,i y y,1 y,d
disjoint from P∗.
1
• If y is a vertex of P∗ and x∈Ω , then
y
(1+ε)−1kyk−d−1≤kxk−d−1 ≤(1+ε)kyk−d−1.
• If y is a vertex of P∗, then (1+ε)σ(Sd−1)−1kyk−d−1V(Ω )<1.
y
Let y be a vertex of P∗, which we keep fixed until (29). For p=0,...,d, and a
(d−1)-dimensional convex set F, we define
1 if P has a supporting hyperplane at y that is parallel to affF, and
θp = affF separates y from K∗ and from p points out of w ,...,w ,
y,F 1 y,1 y,d
0 otherwise.
We write Φ to denote the linear map with Φ e = w −y for i = 1,...,d, and
y y i y,i
hence detΦ =d!V(Ω ). Let
y y
Tp (P∗)= µ∗(S )qθp .
q,y n P F y,F
F∈XF(Pn∗)
10 KA´ROLYJ.BO¨RO¨CZKYANDROLFSCHNEIDER
A standard argument yields
ETp (P∗)= n µ∗(S )q 1−µ∗(C ) n−d
q,y n d P [x1,...,xd] P [x1,...,xd]
(cid:18) (cid:19)ZXPd
(23) × θp dµ∗(x )···(cid:0)dµ∗(x ), (cid:1)
y,[x1,...,xd] P 1 P d
whereC denotesthepartofP∗containingythatiscutoffbyaff{x ,...,x }.
[x1,...,xd] 1 d
We start with the case p = d. It follows by the definition of ω that, writing
β :=σ(Sd−1)kykd+1, we have
n
(24) ETd (P∗)≥ (1+ε)−(d+q)β−(d+q) V(S )q
q,y n d [x1,...,xd]
(cid:18) (cid:19) ZΩdy
(1+ε)V(C ) n−d
(25) × 1− [x1,...,xd] θd dx ···dx .
β y,[x1,...,xd] 1 d
(cid:18) (cid:19)
We write Id to denote the integral in (25). Applying Φ−1, we deduce that
n−d
d!V(Ω )(1+ε)V(C )
Id =(d!V(Ω ))q+d V(S )q 1− y [x1,...,xd]
y ZΩed [x1,...,xd] β !
e
× θ˜d dx ···dxe.
[x1,...,xd] 1 d
Following [2, p. 298–299],we apply first a Blaschke–Petkantschinformula and the
definition of M (·), then (19) and (20), to obtain
q
Id =(d!V(Ω ))q+dd−qM (∆ )(d−1)!
y q+1 d−1
n−d
d!V(Ω )(1+ε)V(C )
× η˜q λd+q+1(H ∩Ω) 1− y H θ˜d dµ(H)
H d−1 β H
ZH !
e
=(d!V(Ω ))q+dd−qM e(∆ )(d−1)!−(d+q)
y q+1 d−1
n−d
V(Ω )(1+ε)
× (s ···s )d+q−1 1− y s ···s d(s ,...,s ).
1 d 1 d 1 d
β
Z(0,1]d (cid:18) (cid:19)
We may apply (21) because V(Ω )(1+ε)/β <1 by the choice of ω. We deduce by
y
(24) that
n
ETd (P∗)≥ (1+ε)−(d+q)β−(d+q)·Id
q,y n d
(cid:18) (cid:19)
(d+q−1)!dd−1M (∆ ) lnd−1n lnd−2n
q+1 d−1
(26) = −Γ .
(d−1)!2(1+ε)2d+2q nq 1 nq
Now a similar argument with the obvious changes leads to
(27)
(1+ε)2d+2q(d+q−1)!dd−1M (∆ ) lnd−1n lnd−2n
ETd (P∗)≤ q+1 d−1 +Γ .
q,y n (d−1)!2 nq 2 nq
Next, let p ∈ {1,...,d−1}. We define τ to be the smallest number such that
τ−1 ≤ σ(Sd−1)kxkd+1 ≤ τ for x ∈ X , and set P := Φ−1(P − y). Starting
P y y
from(23), applying Φ−1 and then the Blaschke–Petkantschinformula and(16), we
y
