R E S E A R C H
Open Access
The
i
th
p
-geominimal surface area
Tongyi Ma
**Correspondence:
[email protected]; [email protected] School of Mathematics and Statistics, Hexi University, Zhangye, Gansu 734000, P.R. China
Abstract
In this paper, we introduce the concept ofithp-geominimal surface area, which extends the notion ofp-geominimal surface area by Lutwak. Further, we prove some of its properties and related inequalities for this new notion.
MSC: 52A30; 52A40
Keywords: convex bodies;ithp-geominimal surface area;p-geominimal area ratio; Brunn-Minkowski-Firey theory
1 Introduction
During the past three decades, the investigations of the classical affine surface area have received great attention from the articles [–] or books [, ]. Based on the classical affine surface area, Lutwak [] introduced the notion ofp-affine surface area and obtained some isoperimetric inequalities forp-affine surface area.
Geominimal surface area was introduced by Petty [] more than three decades ago. As Petty stated, this concept serves as a bridge connecting affine differential geometry, relative differential geometry, and Minkowskian geometry. Based on the classical geominimal sur-face area, Lutwak [] introduced the notion ofp-geominimal surface area and obtained some inequalities for it. Regarding the studies ofp-affine surface area andp-geominimal surface area as well as its dual object, also see [, –].
LetCndenote the set of compact convex subsets of the Euclideann-spaceRn. The subset ofCnconsisting of convex bodies (compact, convex sets with non-empty interiors) will be
denoted byKn. For the set of convex bodies containing the origin in their interiors, write Kn
o, and letKcndenote the set of convex bodies whose centroid lies at the origin. As usual, Sn–denotes the unit sphere with unit ballBn,ω
nthe volume ofBn.
Petty [] defined the geominimal surface areaG(K) of a bodyK∈Kn oby
ω
n
nG(K) =inf
nV(K,Q)V
Q∗n:Q∈Kn o
. (.)
Forp≥ andK∈Kon, Lutwak [] defined thep-geominimal surface areaGp(K) ofK
by
ω
p n
nGp(K) =inf
nVp(K,Q)VQ∗
p
n:Q∈Kn o
. (.)
Moreover, Lutwak proved the following inequalities for thep-geominimal surface area.
Theorem . Let K∈Kn
c and p≥,then
Gp(K)n≤nnωnpV(K)n–p, (.)
with equality if and only if K is an ellipsoid.
LetFondenote the subset ofKnowhich has a positive continuous function, and letp(K)
denote thep-affine surface area ofK.
Theorem . Let K∈Fonand p≥,then
p(K)n+p≤(nωn)pGp(K)n, (.)
with equality if and only if K is of p-elliptic type.
Theorem . If K∈Kn
oand≤p≤q,then
Gp(K)n nnV(K)n–p
p
≤
Gq(K)n nnV(K)n–q
q
, (.)
with equality if and only if K is p-selfminimal.
The purpose of this paper is to further extend Lutwak’sp-geominimal surface area to the
ithp-geominimal surface area. The technique we will use is that of the method designed by Lutwak []. Now, we define the notion ofithp-geominimal surface area as follows:
ω
p n–i
n Gp,i(K) =inf
nWp,i(K,Q)Wi
Q∗
p
n–i:Q∈Kn
o
, (.)
wherei∈ {, , . . . ,n– }.
The main results are stated as follows. First, we establish the extended versions of The-orems ., . and . given by TheThe-orems ., . and ..
Theorem . If p≥and i∈ {, , . . . ,n– },and K∈Kn o,then
Gp,i(K)n–i≤nn–i
ω–nωiωn–ip
n i
–p
Wi(K)n–p–i, (.)
with equality for i= if and only if K is an ellipsoid,for≤i<n if and only if all(n–i)
-dimensional convex bodies which are contained in K are balls.
Ifi= , (.) is just inequality (.).
Let(pi)(K) denote the (i, )-typep-affine surface area ofK(see Section .).
Theorem . If p≥,i∈ {, , . . . ,n– },and K∈Fn i,o,then
(pi)(K)n+p–i≤(nωn)pGp,i(K)n–i, (.)
Theorem . If K∈Kn
oand i∈ {, , . . . ,n– },then for≤p≤q,
Gp,i(K)n–i nn–iWi(K)n–p–i
/p ≤
Gq,i(K)n–i nn–iWi(K)n–q–i
/q
, (.)
with equality if and only if K is ith p-selfminimal.
The proofs of Theorems .-. will be given in Section of this paper. Moreover, in Section we also establish some properties of theithp-geominimal surface area which may be required in the proofs of main results.
2 Background material for Brunn-Minkowski-Firey theory
2.1 Support function, radial function and polar of a convex body
Forφ∈GL(n), letφt,φ–, andφ–tdenote the transpose, inverse, and inverse of the trans-pose ofφ. ForK∈Kn, leth(K,·) :Rn→(–∞,∞) denote the support function ofK∈Kn, i.e., forx∈Rn,
h(K,x) =max{x·y:y∈K},
whereu·xdenotes the standard inner product ofuandx. Forφ∈GL(n), then obviously
h(φK,x) =h(K,φtx). For the sake of convenience, we writehKrather thanh(K,·) for the support function ofK. Apparently, forK,L∈Kn,K⊆Lif and only ifhK≤hL. The setKn will be viewed as equipped with the Hausdorff metricddefined byδ(K,L) =|hK–hL|∞,
where| · |∞ is thesup(ormax) norm on the space of continuous functions on the unit sphereC(Sn–).
For a compact subsetLofRn, which is star-shaped with respect to the origin, we shall
useρ(L,·) to denote its radial function;i.e., foru∈Sn–,
ρ(L,u) =ρL(u) =max{λ> :λu∈L}.
Ifρ(L,·) is continuous and positive,Lwill be called a star body, andSn
o will be used to
denote the class of star bodies inRncontaining the origin in their interiors. Apparently, for
K,L∈Sn,K⊆Lif and only ifρ
K≤ρL. Two star bodiesKandLare said to be dilates (of
one another) ifρ(K,u)/ρ(L,u) is independent ofu∈Sn–. Letδdenote the radial Hausdorff metric as follows: ifK,L∈Sn
o, thenδ(K,L) =|ρK–ρL|∞.
ForK∈Kn
o, the polar bodyK∗ofKis defined by
K∗=x∈Rn:x·y≤,y∈K.
Obviously, we have (K∗)∗=K. ForK∈Kno, the support and radial functions of the polar bodyK∗ofKare defined respectively by (see [, ])
hK∗(u) =
ρK(u)
and ρK∗(u) =
hK(u)
Define the Santaló product ofK∈Kn
o byV(K)V(K∗). The Blaschke-Santaló inequality
(see [, ]) is one of the fundamental affine isoperimetric inequalities. It states that if
K∈Kn
c, then
V(K)VK∗≤ωn, (.)
with equality if and only ifKis an ellipsoid.
2.2 The mixedp-quermassintegrals and dual mixedp-quermassintegrals
ForK∈Knandi∈ {, , . . . ,n– }, the quermassintegralsW
i(K) ofKare defined by (see
[])
Wi(K) =
n Sn–
h(K,u)dSi(K,u). (.)
From (.), we easily see thatW(K) =V(K).
In the literature they have two representations, the quermassintegralsWi(K) and the
intrinsic volumesVi(K), and we shall use both throughout. They are defined by
V(K , . . . ,K n–i
,Bn , . . . ,Bn i
) =Wi(K) = ωni
i
Vn–i(K), i= , , . . . ,n. (.)
For real p≥,K,L∈Kno, andα,β≥ (not both zero), the Firey linear combination
α·K+pβ·Lis defined by (see [])
h(α·K+pβ·L,·)p=αh(K,·)p+αh(L,·)p.
Note that ‘·’ rather than ‘·p’ is written for Firey scalar multiplication.
ForK,L∈Kn
o,ε> , and realp≥, the mixedp-quermassintegralsWp,i(K,L) ofKand L,i∈ {, , . . . ,n– }, are defined by (see [])
n–i
p Wp,i(K,L) =εlim→+
Wi(K+pε·L) –Wi(K)
ε .
Obviously, forp= ,W,i(K,L) is just the classical mixed quermassintegrals Wi(K,L).
Fori= , the mixedp-quermassintegralsWp,(K,L) are just thep-mixed volumeVp(K,L). Forp≥,i∈ {, , . . . ,n– } and eachK∈Kn
o, there exists a positive Borel measure Sp,i(K,·) on Sn–such that the mixedp-quermassintegralsWp,i(K,L) have the following
integral representation (see []):
Wp,i(K,L) =
n Sn–h p
L(v)dSp,i(K,v) (.)
for allL∈Kn
o. It turns out that the measureSp,i(K,·),i∈ {, , . . . ,n– }, onSn–is absolutely
continuous with respect toSi(K,·), and has the Radon-Nikodym derivative
dSp,i(K,·) dSi(K,·)
=h–p(K,·). (.)
Together with (.) and (.), forK∈Kn
An immediate consequence of the definition of Firey linear combination, and the in-tegral representation (.), is that forQ∈Kn
o, the mixedp-quermassintegralsWp,i(Q,·) : Kn
o→(,∞) are Firey linear.
Proposition . Suppose K,L,Q∈Kn
oandλ,μ≥.If p≥,i∈ {, , . . . ,n– },then
Wp,i(Q,λ·K+pμ·L) =λWp,i(Q,K) +μWp,i(Q,L). (.)
It will be helpful to introduce the following notation. Define the inner radiusr(K) and the outer radiusR(K) ofK∈Kn
oby
r(K) =max{λ> :λBn⊂K} and R(K) =min{λ> :K⊂λBn}.
Recall thatBnis the unit ball centered at the origin. Thus,
r(K) = min
u∈Sn–hK(u) and R(K) =umax∈Sn–hK(u).
Obviously, the bodyKis contained in the closure of the annulusR(K)Bn\r(K)Bn. Note that the notions of inner and outer radii as defined here are not translation invariant.
The next proposition shows that the functional Wp,i(K,·)/p :Kno →(,∞) is
Lips-chitzian. This observation will be needed in Sections and .
Proposition . If K,L,Q∈Knoand p≥,i∈ {, , . . . ,n– },then
Wp,i(Q,K)/p–Wp,i(Q,L)/p≤ |hK–hL|∞Wi(Q)/p/r(Q). (.)
Proof The Minkowski integral inequality, together with (.) and (.), gives
Wp,i(Q,K)/p≤Wp,i(Q,L)/p+
n Sn–
hK(u) –hL(u) p
h(Q,u)–pdSi(Q,u)
/p
≤Wp,i(Q,L)/p+|hK–hL|∞Wi(Q)/p max u∈Sn–hQ(u)
–.
ForK∈Sn
oand any reali, theith dual quermassintegralsWi(K) ofKare defined by (see
[, ])
Wi(K) =
n Sn–
ρKn–i(u)dS(u). (.)
Obviously,W(K) =V(K).
An immediate consequence of the definition ofith dual quermassintegrals is as follows.
Proposition . If p≥and i∈Rn,then the functionalW
i(·) :Son→(,∞)is continuous.
ForK,L∈Sn
o,p≥ andλ,μ≥ (not both zero), thep-harmonic radial combination
λ♦K+ˆ–pμ♦L∈Sonis defined by (see [])
IfK,L∈Kn
o(rather than being inSon), then
λ♦K+ˆ–pμ♦L=
λ·K∗+pμ·L∗
∗
.
ForK,L∈Sn
o,ε> ,p≥, and reali=n, the dual mixedp-quermassintegralsW–p,i(K,L)
ofKandLare defined by (see [])
n–i
–p W–p,i(K,L) =εlim→+
Wi(K+ˆ–pε♦L) –Wi(K)
ε . (.)
Ifi= , we easily see that (.) is just the definition of dualp-mixed volume,i.e.,W–p,(K,
L) =V–p(K,L).
From (.), the integral representation of the dual mixedp-quermassintegrals is given by Wang and Leng []: IfK,L∈Sn
o,p≥, and reali=n,i=n+p, then
W–p,i(K,L) =
n Sn–
ρKn+p–i(u)ρL–p(u)dS(u). (.)
Together with (.) and (.), forK∈Sn
o,p≥, andi=n,n+p, it follows thatW–p,i(K,K) =
Wi(K).
Further, Wang and Leng [] proved the following analog of the Minkowski inequality for the dual mixedp-quermassintegrals.
Lemma . If K,L∈Sn
o,p≥,then for i<n or i>n+p,
W–p,i(K,L)n–i≥Wi(K)n+p–iWi(L)–p, (.) with equality in every inequality if and only if K and L are dilates of each other.For n<i<
n+p,inequality(.)is reverse.
Another consequence of Lemma . will be needed.
Lemma .([]) Suppose K,L∈Sn
o,p≥andλ,μ> .If real i<n or n<i<n+p,then
Wi(λ♦K+ˆ–pμ♦L)–p/(n–i)≥λWi(K)–p/(n–i)+μWi(L)–p/(n–i), (.) with equality in every inequality if and only if K and L are dilates of each other.For i>n+p,
inequality(.)is reverse.
2.3 Theithp-curvature function andithp-curvature image
A convex bodyK∈Knis said to have a continuousith curvature functionf
i(K,·) :Sn–→
Rif its mixed surface area measureSi(K,·) is absolutely continuous with respect to the
spherical Lebesgue measureSand has the Radon-Nikodym derivative (see [])
dSi(K,·)
dS =fi(K,·) fori∈ {, , . . . ,n– }. (.)
LetFin,Fin,o,Fin,cdenote the sets of all bodies inKn,Kno,Knc, respectively, that have an
ith positive continuous curvature function. In particular,Fn:=Fn,Fn
If∂Kis a regularC-hypersurface with (everywhere) positive principal curvatures, then
K∈Fn
i for alli, and the curvature functions ofKare proportional to the elementary
sym-metric functions of the principal radii of curvature (viewed as functions of the outer nor-mals) ofK. Thus,f(K,u) is the reciprocal Gauss curvature of∂Kat the point of∂Kwhose outer normal isu, whilefn–(K,u) is proportional to the arithmetic mean of the radii of curvature of∂Kat the point whose outer normal isu.
A convex bodyK∈Kn
o is said to have ap-curvature functionfp(K,·) :Sn–→Rif its p-surface area measure Sp(K,·) is absolutely continuous with respect to the spherical Lebesgue measureSand has the Radon-Nikodym derivative (see [])
dSp(K,·)
dS =fp(K,·). (.)
Lutwak [] showed the notion ofp-curvature image as follows: For eachK∈Fonand
p≥, definepK∈Sn
o, thep-curvature image ofK, by
ρ(pK,·)n+p=
V(pK)
ωn
fp(K,·). (.)
Note that forp= , this definition is different from the classical curvature image (see []). Recently, Liu et al.[], Lu and Wang [] as well as Ma and Liu [, ] indepen-dently introduced the concept ofithp-curvature function ofK∈Kn
oas follows: Letp≥
andi∈ {, , . . . ,n– }, a convex body K∈Kn
o is said to have an ithp-curvature
func-tionfp,i(K,·) :Sn–→Rif itsithp-surface area measureSp,i(K,·) is absolutely continuous
with respect to the spherical Lebesgue measure Sand has the Radon-Nikodym deriva-tive
dSp,i(K,·)
dS =fp,i(K,·). (.)
If theith surface area measureSi(K,·) is absolutely continuous with respect to the spher-ical Lebesgue measureS, we have
fp,i(K,·) =h(K,·)–pfi(K,·). (.)
According to the concept ofithp-curvature function of a convex body, Lu and Wang [] and Ma [] introduced independently the concept ofithp-curvature image of a convex body as follows: For eachK∈Fin,o,i∈ {, , . . . ,n– }and realp≥, definep,iK∈Son, the ithp-curvature image ofK, by
ρ(p,iK,·)n+p–i=
Wi(p,iK)
ωn
fp,i(K,·). (.)
The unusual normalization of definition (.) is chosen so that, for the unit ball Bn, it follows thatp,iBn=Bn. From definitions (.), (.) and formula (.), ifi= , then
p,K=pK.
Proposition . If p≥,i∈ {, , . . . ,n– }and K∈Fin,o,then
Wp,i
K,Q∗=ωnW–p,i(p,iK,Q)/Wi(p,iK) (.)
for all Q∈Sn o.
Recently, Ma [] introduced the concept of (i, )-typep-affine surface area as follows: Letp≥ andi∈ {, , . . . ,n– }, the (i, )-typep-affine surface area(pi)(K) ofK∈Fin,ois
defined by
(pi)(K) =
Sn–fp,i(K,u)
n–i n+p–idS(u).
An immediate consequence of the definition ofithp-curvature image and the integral representations of(pi)andWi is the following proposition.
Proposition . If p≥,i∈ {, , . . . ,n– },and K∈Fn i,o,then
(pi)(K) =nω
n–i n+p–i
n Wi(p,iK) p
n+p–i. (.)
A bodyK∈Fn
o is ofithp-elliptic type if the functionfp,i(K,·)/(n+p–i)is the support
func-tion of a convex body inKno,i.e.,K is ofithp-elliptic type if there exists a bodyQ∈Kon
such that
fp,i(K,·) =h(Q,·)–(n+p–i).
Define
Wn p,i=
K∈Fin,o: there existsQ∈Knowithfp,i(K,·) =h(Q,·)–(n+p–i)
.
An immediate consequence of the definition ofWpn,iand the definition ofp,iis the
fol-lowing.
Proposition . If p≥,i∈ {, , . . . ,n– },and K∈Fn i,o,then
K∈Wpn,i if and only if p,iK∈Kno.
3 Theithp-geominimal surface area
LetO(n) denote an orthogonal transformation group inRn. We will give the following lemmas and propositions.
Lemma .([]) Suppose K,L∈Kn
o,p≥and i∈ {, , . . . ,n– },then for anyφ∈O(n),
Wp,i(φK,φL) =Wp,i(K,L).
Lemma .([]) Suppose K,L∈Son,p≥and real i∈Ras well as i=n,i=n+p,then,
for anyφ∈O(n),
Specifically,
Wi(φK) =Wi(K) for anyφ∈O(n).
An immediate consequence of the definition ofGp,iand Lemma . and Lemma . is
the following.
Proposition . Suppose K∈Kno.If p≥,i∈ {, , . . .n– }andφ∈O(n),then
Gp,i(φK) =Gp,i(K).
Lemma . If p≥,and Kjis a sequence of bodies inKnosuch that Kj→K∈Kon,then for i∈ {, , . . . ,n– },Sp,i(Kj,·)→Sp,i(K,·)weakly.
Proof Supposef ∈C(Sn–). SinceKj→K
, by the definition of support function,hKj →
hK uniformly onSn–. Since the continuous functionhK is positive,hKj are uniformly bounded away from . It follows thath–Kp
j →h –p
K uniformly onS
n–, and thus
fh–Kp
j →fh –p
K uniformly onS n–.
ButKj→Kalso implies that
Si(Kj,·)→Si(K,·) weakly onSn–
follows from the weak continuity of surface area measures (see, for example, Schneider [, ]). Hence,
Sn–f(u)h(Kj,u)
–pdSi(Kj,u)→
Sn–f(u)h(K,u)
–pdSi(K
,u),
or equivalently,
Sn–f(u)dSp,i(Kj,u)→ Sn–f(u)dSp,i(K,u).
Lemma . Suppose Kj→K∈Knand Lj→L∈Kn.If p≥and i∈ {, , . . . ,n– },
then Wp,i(Kj,Lj)→Wp,i(K,L).
Proof SincehLj →hL uniformly onS
n–, and hL is continuous, the hL
i are uniformly bounded onSn–. Hence,
hpLj→hpL uniformly onSn–.
By Lemma .,Kj→Kimplies that
Sp,i(Kj,·)→Sp,i(K,·) weakly onSn–.
Hence,
Sn–h p
LjdSp,i(Kj,u)→
Sn–h p
By the definition of dual mixed p-quermassintegrals and the continuity of the radial function, we have the following.
Lemma . Suppose Kj→K∈Snand Lj→L∈Sn.If p≥,i∈R,and i=n,i=n+p,
thenW–p,i(Kj,Lj)→W–p,i(K,L).
An immediate consequence of the definition of(pi) and Lemma . and Lemma . is
the following.
Proposition . For p≥and i∈ {, , . . .n– },the function Gp,i:Kno→(,∞)is upper
semicontinuous.
SupposeK∈Kn
oandL∈Son. Forp≥, defineWp,i(K,L∗) by
Wp,i
K,L∗=
n Sn–ρL(u)
–pdSp
,i(K,u).
SincehQ∗ = /ρQforQ∈Kno, it follows from the integral representation (.) that, ifL
happens to belong toKon(rather than just toSon), the new definition ofWp,i(K,L∗) agrees
with the old definition.
An immediate consequence of Lemma . is as follows.
Lemma . If p≥and L∈Sn
o,then Wp,i(·,L∗) :Kno→(,∞)is continuous.
The following simple fact will be needed.
Proposition . Suppose Kj∈Kn
o,Kj→L∈Cnand≤i≤n.If the sequenceWi(Kj∗)is bounded,then L∈Kon.
Proof Note that forA∈Sonand <i<n,i∈R, it follows thatWi(A)≤V(A)(n–i)/nωin/nwith equality if and only ifAis ann-ball centered at the origin (see []). We chose two non-negative real numbersc,csuch thatcωni/nV(Kj∗)(n–i)/n≤Wi(Kj∗)≤cfor allj. SinceLis compact, there exists a realrsuch thatL⊂rBn, and sinceKj→L, the numberrmay be chosen so thatKj⊂rBnfor alli, as well. (Recall thatBnis the unit ball centered at the origin.)
For eachj, let
rj=r(Kj) = min
u∈Sn–h(Kj,u) =h(Kj,uj),
whereuj∈Sn–is any point where this minimum is attained. Sinceρ(Kj∗,uj) = /h(Kj,uj) = r–j , it follows thatKj∗contains the pointrj–ui. SinceKj⊂rBn, it follows thatρ(Kj∗,uj) =
/h(Kj,uj)≥/r.
Thus, Kj∗ contains the right cone whose apex is r–j ui and whose base is an (n– )-dimensional ball of radiusr–
that lies in the subspace orthogonal touj. Thus, for ≤i≤n,
ωn–
n r
–n
r–j ≤V
Kj∗≤c–
n n–i ω
–n–ii
n Wi
Kj∗
n n–i≤c
c–
n n–iω–
i n–i
and hencerj≥nωn–ω i n–i
n r–n(cc– )– n
n–i. Hence the ball, centered at the origin, of radius
nωn–ω
i n–i
n r–n(cc– )– n
n–i is contained in eachKj, and thus this ball is contained inLas
well.
ForK∈Kn
o, letPp,iKdenote the compact convex set whose support function, forx∈Rn, is given by
h(Pp,iK,x)p=
n Sn–
–p|x·u|+x·up
dSp,i(K,u).
The fact that the functionh(Pp,iK,·) is convex and hence is the support function of a
compact convex set is a direct consequence of the Minkowski integral inequality. Obvi-ously,h(Pp,iK,·)≥ onSn–. Thath(Pp,iK,·) > onSn–follows from the fact that the
sur-face area measure of a convex body cannot be concentrated on a closed hemisphere ofSn–. If it were the case thath(Pp,iK,·) = , thenSp,i(K,·), and thus theith surface area measure Si(K,·) would be concentrated on a closed hemisphere bounded by the great sphere ofSn– that is orthogonal tou. Sinceh(Pp,i,·) is positive,Pp,iK∈Kon.
ForK∈Kn
oandu∈Sn–, letwi(K|u⊥) denote the (n– )-dimensional quermassintegrals
ofK|u⊥, the image of the orthogonal projection ofKonto the (n–)-dimensional subspace ofRnthat is orthogonal tou. Theith projection bodyiK∈Kn
oofK∈Kon,i= , , . . . ,n–,
is the body whose support function is given by
h(iK,u) =wiK|u⊥= Sn–
u·udSiK,u
foru∈Sn–. (See the survey of Lutwak [].) Since forK∈Kno,
Sn–
u dSi(K,u) = ,
it follows from (.) that
nP,iK=iK.
As noted previously,h(Pp,iK,·) > onSn–. However a slightly stronger statement will be needed in this section.
Lemma . For p≥,i∈ {, , . . .n– },and K∈Kn o,then
h(Pp,iK,u)≥Wi(K)(–p)/pwi
K|u⊥/n
for all u∈Sn–.
Proof Since from (.),
h(Pp,iK,x)p=
n Sn–
|
x·u|+x·u
h(K,u)
p
and
Wi(K) =
n Sn–
h(K,u)dSi(K,u),
it follows from Jensen’s inequality that for allu∈Sn–,
Wi(K)–/ph(Pp,iK,u)≥Wi(K)–h(P,iK,u).
To complete the proof, recall thatnh(P,iK,u) =h(iK,u) =wi(K|u⊥).
Proposition . If p≥,i∈ {, , . . . ,n– },and K∈Kn
o,there exists a unique body K∈ Kn
osuch that
Gp,i(K) =nWp,i(K,K) and Wi
K∗=ωn.
Proof Choosecsuch thath(Pp,iK,·)p≥c> onSn–. From the definition ofGp,i(K), there
exists a sequenceMj∈Knosuch thatWi(Mj∗) =ωnwithWp,i(K,Bn)≥Wp,i(K,Mj) for allj,
and
nWp,i(K,Mj)→Gp,i(K).
To see thatMj∈Kn
oare uniformly bounded, let
Rj=R(Mj) =ρ(Mj,uj) =max
ρ(Mj,u) :u∈Sn–
,
whereujis any of the points inSn–at which this maximum is attained.
Since the support function of Mj dominates that of the convex set ej={λuj: ≤
λ≤Rj} ⊂Mj, and since the measure Sp,i(K,·) is positive, it follows thatWp,i(K,Mj)≥ Rpjh(Pp,iK,uj)p. Hence,
Wp,i(K,Bn)≥Wp,i(K,Mj)≥Rpjh(Pp,iK,uj)p≥Rpjc.
SinceMjare uniformly bounded, the Blaschke selection theorem guarantees the
exis-tence of a subsequence ofMj, which will also be denoted byMj, and a compact convex L∈Cn, such thatMj→L. SinceWi(Mj∗) =ωn, Proposition . givesL∈Kno. Now,Mj→L
implies thatM∗j →L∗, and sinceWi(Mj∗) =ωn, it follows thatWi(L∗) =ωn. Lemma . can
now be used to conclude thatLwill serve as the desired bodyK.
The uniqueness of the minimizing body is easily demonstrated as follows. Suppose
L,L∈Kno, such thatWi(L∗) =ωn=Wi(L∗), and
Wp,i(K,L)Wi
L∗p/(n–i)=infWp,i(K,Q)Wi
Q∗p/(n–i):Q∈Kno
=Wp,i(K,L)Wi
L∗p/(n–i). DefineL∈Kn
oby
L=
Proposition . shows that
Wp,i(K,L) =Wp,i(K,L) =Wp,i(K,L).
Since, obviously,
L∗= ♦L
∗
+ˆp
♦L
∗
andWi(L∗) =ωn=Wi(L∗), it follows from Lemma . that
WiL∗≤ωn
with equality if and only ifL=L. Thus,
Wp,i(K,L)Wi
L∗p/(n–i)<Wp,i(K,L)Wi
L∗p/(n–i)
is the contradiction that would arise if it were the case thatL=L.
The unique body whose existence is guaranteed by Proposition . will be denoted by Tp,iK and will be called the ith p-Petty body of K. The polar body of Tp,iK will be denoted byTp∗,iK rather than (Tp,iK)∗. Whenp= , the subscript will often be sup-pressed. Thus, for K ∈Kn
o, p≥, and i∈ {, , . . . ,n– }, the body Tp,iK is defined
by
Gp,i(K) =nWp,i(K,Tp,iK) and Wi
Tp∗,i=ωn.
The next proposition shows that the mappingTp,i:Kon→Konis an orthogonal
transfor-mation invariant mapping.
Proposition . If p≥,i∈ {, , . . . ,n– },and K∈Kn
o,then forφ∈O(n),
Tp,iφK=φTp,iK.
Proof From the definition ofTp,iand Proposition .,
nWp,i(K,Tp,iK) =Gp,i(K) =Gp,i(φK) =nWp,i(φK,Tp,iφK).
By Lemma .,
Wp,i(K,Tp,iK) =Wp,i(φK,Tp,iφK) =Wp,i
K,φ–Tp,iφK
.
The uniqueness part of Proposition . shows thatTp,iK=φ–Tp,iφK, which is the desired
result.
Lemma .([]) If K∈Knand i∈ {, , . . . ,n– },then
Wi(K)≥ω
i n
nV(K)
n–i n ,
with equality if and only if K is an n-ball.
The following crude bound on the size ofTp,iKwill be helpful.
Lemma . Suppose p≥,i∈ {, , . . . ,n– },and K∈Kn
o.If r,R> are such that
rBn⊂K⊂RBn,
then
h(Tp,iK,u)≤
nωiωn–i n
i
ωn– (R/r)n–i
for all u∈Sn–.
Proof From the integral representation (.) and formulas (.) and (.), the trivial esti-mate follows:
Wp,i(K,Bn)≤
Wi(K) minu∈Sn–hpK(u)
=n ωiVn–i(K)
i
minu∈Sn–hpK(u)≤
ωiωn–iRn–i n
i
rp .
From the minimality property ofTp,iK, it follows that
Wp,i(K,Tp,iK)≤Wp,i(K,Bn).
Letube any point inSn–such that
ρ(Tp,iK,u) =max
ρ(Tp,iK,u) :u∈Sn–
=R(Tp,iK).
Since the support function ofTp,iKdominates that of the convex sete={λu: ≤λ≤
R(Tp,iK)} ⊂Tp,iK, it follows that
R(Tp,iK)ph(Pp,iK,u)p≤Wp,i(K,Tp,iK).
But from Lemma ., Lemma . and formula (.), it follows that
h(Pp,iK,u)p≥Wi(K)–pwi
K|u⊥p/np
=n–p
n i
p–
ωi–pVn–i(K)–pwi
K|u⊥p
≥n–p
n i
p–
ωi–pVn–i(K)–pω
ip n– n–v
K|u⊥ (n––i)p
n–
≥n–p
n i
p–
ωi–pω
ip n– n–
Rn–iωn–i
–p rn–ωn–
(n––i)p n– .
If the outer radii of a sequence of bodies are uniformly bounded from above and the inner radii of the sequence are bounded away from , then the same is true for the radii of theithp-Petty bodies of the sequence. This is contained in the following lemma.
Lemma . Suppose p≥,i∈ {, , . . . ,n– }.If Kj∈Kn
o is a family of bodies for which there exist r,R> such that
rBn⊂Kj⊂RBn for all j,
then there exist r,R> such that
rBn⊂Tp,iKj⊂RBn for all j.
Proof The existence ofR> , and thus the fact thatTp,iKjare uniformly bounded, is con-tained in Lemma .. Letrj=r(Tp,iKj) denote the inner radius ofTp,iKj. Thus,
rj= min
u∈Sn–h(Tp,iKj,u) =h(Tp,iKj,uj),
whereuj∈Sn–is any point where this minimum is attained. Suppose that the infimum of rjis . Thus, there exists a subsequence ofTp,iKj, which will not be relabeled, such that
h(Tp,iKj,uj)→.
The Blaschke selection theorem, in conjunction with Proposition ., demonstrates the existence ofM∈Kn
osuch that for a subsequence ofTp,iKj, which will also not be relabeled,
Tp,iKj→M.
Buth(Tp,iKj)→ and|hTp,iKj –hM|∞→ imply thathM(uj)→, which is impossible
since the continuous functionhMis positive.
The casei= of the following proposition is due to Lutwak []. The proof of this propo-sition is based on the one given by Petty and Lutwak.
Proposition . If p≥and i∈ {, , . . . ,n– },then the functional Gp,i:Kno→(,∞)is continuous.
Proof ThatGp,iis upper semicontinuous follows immediately from Lemma .: Theith
p-geominimal surface area
ωpn/(n–i)Gp,i:Kon→(,∞)
is defined as the infimum of continuous functions
nWp,i
·,Q∗ Wi(Q)p/(n–i):Kno→(,∞)
To see thatGp,iis lower semicontinuous atK∈Kon, letKj∈Konbe a sequence of bodies
such thatKj→KowithGp,i(Kj)→l∈R. It will be shown thatl≥Gp,i(K), and thus
lim infGp,i(Kj)≥Gp,i(K).
By Lemma . theTp,iKjare uniformly bounded. The Blaschke selection theorem, in
conjunction with Proposition ., yields the existence of a body M∈Kn
o and a
subse-quence of Tp,iKj, which will not be relabeled, such thatTp,iKj→MandWi(M∗) =ωn.
Lemma . and the facts thatKj→K andTp,iKj→Mmay be used to conclude that
Gp,i(Kj) =nWp,i(Kj,Tp,iKj)→nWp,i(K,M). NownWp,i(K,M) =l, sinceGp,i(Kj) =l. But
the definition ofGp,i(K) shows that
ωpn/(n–i)l=nWp,i(K,M)Wi
M∗p/(n–i)≥ωnp/(n–i)Gp,i(K)
and completes the argument.
The casei= of the following result is due to Lutwak [].
Proposition . If p≥and i∈ {, , . . . ,n– },then the map Tp,i:Kon→Knois continu-ous.
Proof SupposeKj∈Kno such that Kj→K∈Kon. Let Tp,iKj denote a subsequence of Tp,iKj. SinceK∈Kno, Lemma . shows thatTp,iKjare uniformly bounded. The Blaschke selection theorem, in conjunction with Proposition ., yields the existence of a body
M∈Kno and a subsequence ofTp,iKj, which will not be relabeled, such thatTp,iKj→M andWi(M∗) =ωn. Lemma . and the facts thatKj→KandTp,iKj→Mmay be used
to conclude that Gp,i(Kj) =nWp,i(Kj,Tp,iKj)→nWp,i(K,M). But by Proposition .,
Gp,i(Kj)→Gp,i(K). Hence,Gp,i(K) =nWp,i(K,M), and the uniqueness part of Propo-sition . shows thatTp,iK=M.
Hence, every subsequence of the sequence Tp,iKj has a subsequence converging to
Tp,iK.
4 Theithp-geominimal surface area ratio
In [], Lutwak defined thep-geominimal area ratio ofKby
Gp(K)n nnV(K)n–p
/p
.
ForK∈Kn, we define theithp-geominimal area ratio ofKas
Gp,i(K)n–i nn–iWi(K)n–p–i
/p
,
and define theith Santaló product ofK∈KnobyWi(K)Wi(K∗).
Theithp-geominimal area ratio does not exceed theith Santaló product divided byωn.
To see this, just takeQ=Kin the definition ofithp-geominimal surface area
ω
p n–i
n Gp,i(K) =inf
nWp,i(K,Q)Wi
Q∗
p
n–i:Q∈Kn o
Proposition . If p≥and i∈ {, , . . . ,n– },and K∈Kn o,then
ωn
Gp,i(K)n–i nn–iW
i(K)n–p–i
/p
≤Wi(K)WiK∗.
An immediate consequence of Proposition . is as follows.
Theorem . If p≥and i∈ {, , . . . ,n– },and K∈Kn o,then
Gp,i(K)n–i≤nn–iω–npWi(K)n–iWi
K∗p. (.)
Lemma .([]) If K∈Kn
o and i∈ {, . . . ,n– },then
Wi(K)≤Wi(K), (.)
with equality if and only if K is an n-ball(centered at the origin).
Proof of Theorem. Inequality (.), together with (.), (.) and (.), yields
Gp,i(K)n–i≤nn–iω–npWi(K)n–p–i
Wi(K)WiK∗p ≤nn–iω–npWi(K)n–p–iWi(K)WiK∗p
=nn–iω–npWi(K)n–p–iωip
Vn–i(K)Vn–i
K∗pn
i
p
≤nn–iω–nωiωn–ip
n i
–p
Wi(K)n–p–i.
According to the conditions of equality in inequalities (.) and (.), we know that for
i= equality of inequality (.) holds if and only ifKis an ellipsoid for ≤i<nif and only if all (n–i)-dimensional convex bodies contained inKare balls.
Theorem . If p≥,i∈ {, , . . . ,n– },and K∈Fin,o,then
Gp,i(K)n–i≥nn–iωnn–p–iWi(p,iK)p,
with equality if and only if K∈Wn p,i.
Proof SinceGp,i(K) =nWp,i(K,Tp,iK) andWi(Tp∗,iK) =ωn, Proposition . gives Gp,i(K) =nωnW–p,i
p,iK,Tp∗,iK
/Wi(p,iK).
Apply inequality (.) and get
Gp,i(K)n–i≥nn–iωnn–p–iWi(p,iK)p,
with equality if and only if p,iK andTp∗,iK are dilates of each other. SinceTp,iK∈Kno,
To see that ifK∈Wpn,i, there is equality in the inequality of the theorem, combine Propo-sition . with the definition ofGp,i(K) to get
ωpn/(n–i)Gp,i(K) =inf
nωnW–p,i
p,iK,Q∗ Wi
Q∗p/(n–i)/Wi(p,iK) :Q∈Kon
.
SinceK∈Kn
o, by Proposition .,p,iK∈Kon. Thus,Q=∗p,iKgivesnωnWi(p,iK)p/(n–i)≥
ωpn/(n–i)Gp,i(K) and demonstrates the desired equality in the inequality.
An immediate consequence of Theorem . and Proposition . is Theorem .. Suppose ≤p<qandK∈Kn
owithL∈Son. Since
ρL–ph–Kp=ρL–qhK–qp/qh(Kq–p)/q, the Hölder inequality yields the following.
Proposition . Suppose K∈Kn
o,L∈Sonand i∈ {, , . . . ,n– }.If≤p<q,then
Wp,i(K,L∗) Wi(K)
/p ≤
Wq,i(K,L∗) Wi(K)
/q
,
with equality if and only if there exists c> such thatρL=c/hK almost everywhere with respect to Si(K,·).
Suppose ≤p<q, and K ∈Kn
o with L∈Son. From the integral representation of Wp,i(K,L∗), the easy estimate follows:
Wp,i
K,L∗–Wp,i
K,L∗≤Wp,i
K,L∗max u∈Sn–
ρL(u)hK(u)
p–q
– .
This gives the following proposition.
Proposition . Suppose K∈Kno,L∈Sonand i∈ {, , . . . ,n– }.Then the function defined on[,∞)by
p→Wp,i
K,L∗ is continuous.
The definition ofGp,i(K) ofK∈Kon, by
Gp,i(K) nWi(K)
/p
=infWp,i
K,Q∗/Wi(K)/p:Q∈Kno,Wi(Q) =ωn
,
together with Proposition ., shows that theithp-geominimal area ratios are monotone non-decreasing inp.
Proposition . If K∈Kn
oand i∈ {, , . . . ,n– },then for≤p≤q,
Gp,i(K)n–i nn–iW
i(K)n–p–i
/p ≤
Gq,i(K)n–i nn–iW
i(K)n–q–i
/q
The equality conditions for the inequality of Proposition . are given in Theorem .. Proposition . provides a key step in showing the following.
Proposition . If K∈Kn
o and i∈ {, , . . . ,n– },then the function defined on[,∞)by
p→Gp,i(K)
is continuous.
Proof Proposition . shows that the functionψ: [,∞)→(,∞) defined by
ψ(p) =
Gp,i(K) nWi(K)
/p
is monotone. The continuity ofp→Gp,i(K) will be demonstrated by establishing the
con-tinuity ofψ.
Supposepj→p. By Proposition ., there existTpj,iK∈Knsuch thatWi(Tp∗j,iK) =ωn
and
Gpj,i(K) =nWpj,i(K,Tpj,iK)≤nWpj,i(K,Bn).
First assume thatpj≥pfor allj. From the definition ofithp-geominimal surface area and Proposition ., it follows that
ψ(p) =
Wp,i(K,Tp,iK) Wi(K)
/p
≤
Wp
,i(K,Tpj,iK)
Wi(K)
/p
≤
W
pj,i(K,Tpj,iK) Wi(K)
/pj
=ψ(pj)
≤
W
pj,i(K,Tp,iK) Wi(K)
/pj .
The continuity of the functionp→[Wp,i(K,Tp,iK)/Wi(K)]/pshows that
W
pj,i(K,Tp,iK) Wi(K)
/pj
→
Wp,i(K,Tp,iK) Wi(K)
/p
,
and hence
ψ(pj)→ψ(p).
Now assume thatpj≤pfor allj. Thatψ(pj)→ψ(p) will be proven by showing that every subsequence ofψ(pj) has a subsequence converging toψ(p). Letψ(pj) denote a
Lemma . shows thatTpj,iKare uniformly bounded. Thus, the Blaschke selection the-orem and Proposition . can now be used to deduce the existence of a subsequence of
Tpj,iK, which will also be denoted byTpj,iK, and a bodyK∈Kno, withWi(K
∗
) =ωn, such
thatTpj,iK→K. Obviously,
Wpj,i(K,Tpj,iK)
Wi(K)
/pj –
Wp,i(K,K) Wi(K)
/p
≤
W
pj,i(K,Tpj,iK) Wi(K)
/pj –
W
pj,i(K,K) Wi(K)
/pj
+
Wp
j,i(K,K) Wi(K)
/pj –
Wp,i(K,K) Wi(K)
/p
.
By Proposition . the second term of this sum tends to . By Proposition . the first term in this sum is bounded by
h(Tpj,i,·) –h(K,·)∞/r(K),
and sinceTpj,iK→K, this also tends to . Hence,
ψ(pj)→
Wp,i(K,K) Wi(K)
/p ≥
Wp,i(K,Tp,iK) Wi(K)
/p
=ψ(p),
where the inequality is justified by the definition ofithp-geominimal surface area. But by Proposition .,ψis monotone non-decreasing, and henceψ(pj)→ψ(p).
ForK∈Kn
o, letσ(K)⊂Sn–denote the compact set that is the support of theith surface
area measureSi(K,·) ofK;i.e.,ω=Sn–\σ(K) is the largest open subset ofSn–for which Si(K,ω) = . Letv(K)⊂Sn–denote the set of extreme normal directions of∂K.
Lemma . Suppose K∈Konand c> .If hTp,iK=chK almost everywhere with respect to
Si(K,·),then hTp,iK=chKeverywhere.
Proof SincehTp,iK andchK are continuous, andhTp,iK=chK almost everywhere with re-spect toSi(K,·), it follows thathTp,iK=chKonσ(K). Butv(K)⊂σ(K), and hence
hTp,iK=chK onv(K).
But
Tp,iK⊂
u∈v(K)
x∈Rn:u·x≤hTp,iK(u)
⊂
u∈v(K)
x∈Rn:u·x≤hcK(u)
⊂cK
shows thatK∗⊂cTp∗,iK. SinceWi(Tp∗,iK) =ωn, it follows thatWi(K∗)≤cn–iωnwith equality
We now show that indeed there is equality in this inequality, and hencecK=Tp,iK. First note that since hTp,iK=chK almost everywhere with respect to Si(K,·), it follows that hTp,iK=chK almost everywhere with respect to Sp,i(K,·). Hence, from the integral representation (.) it follows that
Gp,i(K) =nWp,i(K,Tp,iK) =nWp,i(K,cK) =cpnWi(K).
From this and the definition ofithp-geominimal surface area, it follows that
cpnWi(K) =Gp,i(K)≤nWp,i(K,K) Wi
K∗/ωn
p/(n–i)
=nWi(K) WiK∗/ωn
p/(n–i) .
HenceWi(K∗)≥cn–iω
n.
Proposition . states that theithp-geominimal ratio is always dominated by the Santaló product (divided byωn);i.e., forK∈Kno,p≥ andi∈ {, , . . . ,n– },
Gp,i(K)n–i nn–iW
i(K)n–p–i
/p
≤Wi(K)WiK∗/ωn.
The main result of this section is that in the limit (asp→ ∞) these two quantities are equal.
Theorem . If p≥,i∈ {, , . . . ,n– },and K∈Kn o,then
lim p→∞
Gp,i(K)n–i nn–iWi(K)n–p–i
/p
=sup p≥
Gp,i(K)n–i nn–iWi(K)n–p–i
/p
=Wi(K)WiK∗/ωn.
Proof Since the first equality is an immediate consequence of Proposition ., only the
second equality needs to be demonstrated.
Proposition . guarantees the existence ofTp,iK∈Konsuch thatWi(Tp∗,iK) =ωnand Gp,i(K) =nWp,i(K,Tp,iK)≤nWp,i(K,Bn) for allp.
By Lemma . it follows that there existsc> such thatTp,iK⊂cBnfor allp. The Blaschke selection theorem and Proposition . may be used to deduce the existence of a subsequence ofTp,iK, which will also be denoted byTp,iK, and a bodyK∈Kno with
Wi(K∗) =ωnsuch thatTp,iK→Kasp→ ∞. Definer(K,K) =max{λ> :λK⊂K}. Now,
Wp,i(K,Tp,iK) Wi(K)
/p
–
r(K,K)
≤
Wp,i(K,Tp,iK)
Wi(K)
/p
–
Wp,i(K,K)
Wi(K)
/p
+
Wp,i(K,K)
Wi(K)
/p
–
r(K,K)
.
Proposition . shows that the first term in this sum is dominated by
