Available online at www.tjnsa.com
J. Nonlinear Sci. Appl. 9 (2016), 1143–1152
Research Article
Lp-dual mixed geominimal surface areas
Yan Lia , Wang Weidonga,∗, Si Linb
a
Department of Mathematics, China Three Gorges University, Yichang, 443002, P. R. China.
b
Department of Mathematics, Beijing Forestry University, Beijing, 100083, P. R. China.
Communicated by R. Saadati
Abstract
Zhu, Zhou and Xu showed an integral formula of Lp -mixed geominimal surface area by the p-Petty body.
In this paper, we give an integral representation of Lp -dual mixed geominimal surface area and establish
c
several related inequalities. 2016
All rights reserved.
Keywords: Lp -mixed geominimal surface area, Lp -dual mixed geominimal surface area, integral
representation.
2010 MSC: 52A20, 52A40.
1. Introduction
Let Kn denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean
space Rn . For the set of convex bodies containing the origin in their interiors and the set of convex bodies
whose centroid lie at the origin in Rn , we write Kon and Kcn , respectively. Let Son and Scn respectively denote
the set of star bodies (about the origin) and the set of star bodies whose centroid lie at the origin in Rn .
Let S n−1 denote the unit sphere in Rn and V (K) denote the n-dimensional volume of the body K. For the
standard unit ball B in Rn , its volume is written by ωn = V (B).
The notion of Lp -geominimal surface area was given by Lutwak (see [7]). For K ∈ Kon , p ≥ 1, the
Lp -geominimal surface area, Gp (K), of K is defined
p
p
ωnn Gp (K) = inf{nVp (K, L)V (L∗ ) n : L ∈ Kon }.
Here Vp (K, L) denotes Lp -mixed volume of K, L ∈ Kon (see [6, 7]) and L∗ denotes the polar of L. For
the case p = 1, Gp (K) is just classical geominimal surface area which is introduced by Petty ([8]). Some
∗
Corresponding author
Email address: wangwd722@163.com (Wang Weidong )
Received 2015-10-05
Y. Li, W. Weidong, S. Lin, J. Nonlinear Sci. Appl. 9 (2016), 1143–1152
1144
affine isoperimetric inequalities related to the classical and Lp geominimal surface areas can be found in
[1, 4, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17].
In [7], Lutwak gave the infimum in the definition of Lp -geominimal surface area.
Proposition 1.1. For K ∈ Kon and p ≥ 1, there exists a unique body Tp K ∈ T n with
Gp (K) = nVp (K, Tp K).
Here T n = {T ∈ Kn : s(T ) = o, V (T ∗ ) = ωn }, s(T ) denotes the Santaló point of T , the body Tp K is
called the p-Petty body of K and Tp∗ K denotes the polar of Tp K. When p = 1, the subscript will often be
suppressed and defined by Petty (see [8]).
From Proposition 1.1, Zhu, Zhou and Xu (see [18]) obtained the following fact.
Proposition 1.2. For K ∈ Fon and p ≥ 1, there exists a unique convex body Tp K ∈ T n with
Z
hpTp K (u)fp (K, u)dS(u).
Gp (K) =
S n−1
Here hM (·) denotes the support function of M ∈ Kn , fp (K, ·) denotes the Lp -curvature function of
K ∈ Kon and Fon denotes the set of Kon that have a positive continuous Lp -curvature function.
Moreover, Zhu, Zhou and Xu (see [18]) studied the Lp -mixed geominimal surface area. They defined the
Lp -mixed geominimal surface areas as follows:
Definition 1.3. Let Ki ∈ Fon and p ≥ 1, for each i (i = 1, · · · , n), there exists a unique body (Petty body
of Ki ) Ti = Tp Ki ∈ T n with
Z
1
Gp (K1 , · · · , Kn ) =
[gp (K1 , u) · · · gp (Kn , u)] n dS(u).
S n−1
Here gp (Ki , ·) = hpTi (·)fp (Ki , ·).
For the Lp -mixed geominimal surface area, they (see [18]) proved the following results.
Theorem 1.4. If n 6= p > 1, and K1 , . . . , Kn ∈ Fon , then for 1 ≤ m ≤ n,
m m−1
Y
Gp (K1 , . . . , Kn )
≤
Gp (K1 , · · · , Kn−m , Kn−i , · · · , Kn−i ).
|
{z
}
i=0
m
Equality holds if and only if the Kj are dilates of each other for j = n − m + 1, · · · , n. If m = 1 equality
holds trivially.
Theorem 1.5. If K, L ∈ Fon , n 6= p ≥ 1, i, j, k ∈ R and i < j < k, then
Gp,j (K, L)k−i ≤ Gp,i (K, L)k−j Gp,k (K, L)j−i
with equality if and only if K and L are dilates.
Theorem 1.6. If K, L ∈ Fcn , n 6= p ≥ 1, i ∈ R and 0 < i < n, then
Gp,i (K, L)Gp,i (K ∗ , L∗ ) ≤ (nωn )2
with equality if and only if K and L are dilated ellipsoids.
Y. Li, W. Weidong, S. Lin, J. Nonlinear Sci. Appl. 9 (2016), 1143–1152
1145
Here Fcn denotes the set of Kcn that have a positive continuous Lp -curvature function.
Based on the Lp -dual mixed volume, Wang and Qi (see [13]) gave the notion of Lp -dual geominimal
surface area as follows:
e −p (K), of K is defined by
Definition 1.7. For K ∈ Kon and p ≥ 1, the Lp -dual geominimal surface area, G
p
−
e −p (K) = inf{nVe−p (K, L)V (L∗ )− np ; L ∈ Kon }.
ωn n G
Here Ve−p (K, L) denotes Lp -dual mixed volume of K and L, and
Z
1
e
ρn+p (u)ρ−p
V−p (K, L) =
L (u)dS(u),
n S n−1 K
(1.1)
(1.2)
where ρK (·) denotes the radial function of K ∈ Son .
In this paper, we firstly obtain the infimum in the above definition as follows:
e ∈ Te n , such that
Proposition 1.8. If K ∈ Kon and p ≥ 1, then there exists a unique body K
e −p (K) = nVe−p (K, K).
e
G
(1.3)
Here Te n = {Te ∈ Kon : V (Te∗ ) = ωn }.
e −p (K).
By Proposition 1.8 and (1.2), we have the following integral representation of G
e ∈ Te n with
Proposition 1.9. For K ∈ Kon and p ≥ 1, there exists a unique body K
Z
e
e u)−p dS(u).
G−p (K) =
ρ(K, u)n+p ρ(K,
(1.4)
S n−1
Next, corresponding to Definition 1.3, we define integral form of Lp -dual mixed geominimal surface area
by Proposition 1.9.
e i ∈ Te n (i = 1, · · · , n) with
Definition 1.10. For each Ki ∈ Kon and p ≥ 1, there exists a unique body K
Z
e −p (K1 , . . . , Kn ) =
e 1 , u)−p · · · ρ(Kn , u)n+p ρ(K
e n , u)−p ] n1 dS(u).
G
[ρ(K1 , u)n+p ρ(K
(1.5)
S n−1
e i , ·)−p , then G
e −p (K1 , . . . , Kn ) can be written as follows:
Let ge−p (Ki , ·) = ρ(Ki , ·)n+p ρ(K
Z
1
e
G−p (K1 , . . . , Kn ) =
[e
g−p (K1 , u) · · · ge−p (Kn , u)] n dS(u).
(1.6)
S n−1
e −p,i (K, L) =
Let K1 = · · · = Kn−i = K and Kn−i+1 = · · · = Kn = L (i = 0, 1, · · · , n) in (1.6), we denote G
|
{z
}
|
{z
}
n−i
i
e −p (K, · · · , K , L, · · · , L). More general, if i is any real, we define that: for K, L ∈ Kn , p ≥ 1, the ith
G
o
| {z } | {z }
n−i
i
e −p,i (K, L), of K and L by
Lp -dual mixed geominimal surface area, G
Z
n−i
i
e
G−p,i (K, L) =
ge−p (K, u) n ge−p (L, u) n dS(u).
(1.7)
S n−1
e = B. Thus, let L = B in (1.7) and write
From Proposition 1.8, we easily know B
e −p,i (K, B) = G
e −p,i (K),
G
(1.8)
Y. Li, W. Weidong, S. Lin, J. Nonlinear Sci. Appl. 9 (2016), 1143–1152
1146
then (1.7), (1.8) and ρ(B, ·) = 1 yield
Z
e −p,i (K) =
G
S n−1
ge−p (K, u)
n−i
n
dS(u).
(1.9)
Obviously, from (1.6), (1.7) and (1.9), we have
e −p,0 (K) = G
e −p (K),
G
e −p,0 (K, L) = G
e −p (K),
G
(1.10)
e −p,n (K, L) = G
e −p (L).
G
(1.11)
Further, associated with the Lp -dual mixed geominimal surface areas, we give the following dual results
of Theorems 1.4,1.5 and 1.6, respectively.
Theorem 1.11. If K1 , · · · , Kn ∈ Kon , 1 ≤ m ≤ n, then for p ≥ 1,
m m−1
Y
e
e −p (K1 , · · · , Kn−m , Kn−i , · · · , Kn−i )
G−p (K1 , · · · , Kn )
≤
G
{z
}
|
i=0
(1.12)
m
with equality if and only if there exist positive constants c1 , c2 , · · · , cm such that for all u ∈ S n−1 ,
n+p
−p
−p
c1 ρn+p
Kn (u)ρ e (u) = c2 ρKn−1 (u)ρ e
Kn−1
Kn
−p
(u) = · · · = cm ρn+p
Kn−m+1 (u)ρ e
Kn−m+1
(u).
Theorem 1.12. For K, L ∈ Kon , p ≥ 1, i, j, k ∈ R. If i < j < k, then
e −p,k (K, L)j−i
e −p,j (K, L)k−i ≤ G
e −p,i (K, L)k−j G
G
(1.13)
with equality if and only if K and L are dilates of each other.
Theorem 1.13. If K, L ∈ Kcn , p ≥ 1, i ∈ R and 0 ≤ i ≤ n, then
e −p,i (K, L)G
e −p,i (K ∗ , L∗ ) ≤ (nωn )2
G
(1.14)
with equality if and only if K and L are dilated ellipsoids of each other.
2. Notation and Background Materials
If K ∈ Kn , then its support function, hK = h(K, ·) : Rn −→ (−∞, +∞), is defined by (see [3])
h(K, x) = max{x · y : y ∈ K},
x ∈ Rn ,
(2.1)
where x · y denotes the standard inner product of x and y.
If K is a compact star-shaped (with respect to the origin) in Rn , then its radial function, ρK (·) = ρ(K, ·) :
Rn \ {0} → [0, ∞), is defined by (see [3, 10])
ρ(K, u) = max{λ ≥ 0 : λu ∈ K}, u ∈ S n−1 .
(2.2)
If ρK is positive and continuous, K will be called a star body (respect to the origin). Two star bodies K
and L are said to be dilates (of one another) if ρK (u)/ρL (u) is independent of u ∈ S n−1 .
If E is a nonempty subset in Rn , the polar set, E ∗ , of E is defined by (see [3, 10])
E ∗ = {x ∈ Rn : x · y ≤ 1, y ∈ E}.
If K ∈ Kon , it follows that K ∗∗ = K and
ρ(K, u)−1 = h(K ∗ , u), ρ(K ∗ , u)−1 = h(K, u)
(2.3)
Y. Li, W. Weidong, S. Lin, J. Nonlinear Sci. Appl. 9 (2016), 1143–1152
1147
for all u ∈ S n−1 .
For K ∈ Kon and its polar body, the well-known Blaschke-Stantaló inequality can be stated that (see [3]):
If K ∈ Kcn , then
V (K)V (K ∗ ) ≤ ωn2
(2.4)
with equality if and only if K is an ellipsoid centered at the origin.
e −p µ ? L ∈
For K, L ∈ Son , p ≥ 1 and λ, µ ≥ 0 (not both zero), the Lp -harmonic radial combination, λ ? K +
n
So , of K and L is defined by (see [2, 7])
e −p µ ? L, ·)−p = λρ(K, ·)−p + µρ(L, ·)−p ,
ρ(λ ? K +
(2.5)
e −p ’ is called Lp -harmonic radial addition and λ ? K denotes the Lp -harmonic radial
where the operation ’+
scalar multiplication. From (2.5), we can obtain λ ? K = λ
If K, L ∈ Kon (rather than being in Son ), then
− p1
K.
e −p µ ? L)∗ = λ · K ∗ +p µ · L∗ .
(λ ? K +
(2.6)
For the Lp -harmonic radial combinations, Lutwak (see [7]) proved the following dual Lp -BrunnMinkowski inequality.
Lemma 2.1. If K, L ∈ Son , p ≥ 1 and λ, µ ≥ 0 (not both zero), then
e −p µ ? L)
V (λ ? K +
−p
n
≥ λV (K)
−p
n
+ µV (L)
−p
n
(2.7)
with equality if and only if K and L are dilates of each other.
3. Proof of Proposition 1.8
In order to prove Proposition 1.8, we require the following lemmas.
Lemma 3.1 ([7]). Let C n denote the set of compact convex subsets of Euclidean n-space Rn . Suppose
Ki ∈ Kon and Ki → L ∈ C n . If the sequence V (Ki∗ ) is bounded, then L ∈ Kon .
Lemma 3.2. Suppose Ki → K ∈ Kon and Li → L ∈ Kon . If p ≥ 1, then Ve−p (Ki , Li ) → Ve−p (K, L).
Proof. Since Ki → K ∈ Kon and Li → L ∈ Kon , ρKi → ρK and ρLi → ρL uniformly on S n−1 . Notice that
ρK , ρL is continuous, the ρKi and ρLi are uniformly bounded on S n−1 . Hence,
−p
−p
n+p
ρn+p
Ki → ρK , ρLi → ρL .
This yields
Z
n+p
ρ(Ki , u)
S n−1
ρ(Li , u)
−p
Z
du →
[ρ(K, u)n+p ρ(L, u)−p ]dS(u),
S n−1
i.e.
Ve−p (Ki , Li ) → Ve−p (K, L).
e −p (K), there exists a sequence Mi ∈ Kon such that
Proof of Proposition 1.8. From the definition of G
V (Mi∗ ) = ωn , with Vp (K, B) ≥ Vp (K, Mi ), for all i, and
e −p (K)
Ve−p (K, Mi ) → G
Since Mi ∈ Kon , thus Mi are uniformly bounded for all i (see [7]). From this, the Blaschke selection theorem
guarantees the existence of a subsequence of the Mi , which will also be denoted by Mi , and a compact
Y. Li, W. Weidong, S. Lin, J. Nonlinear Sci. Appl. 9 (2016), 1143–1152
1148
convex L ∈ C n , such that Mi → L. Since V (Mi∗ ) = ωn , Lemma 3.1 gives L ∈ Kon . Now, Mi → L implies
that Mi∗ → L∗ , and since V (Mi∗ ) = ωn , it follows that V (L∗ ) = ωn . Lemma 3.2 can now be used to conclude
e
that L will serve as the desired body K.
The uniqueness of the minimizing body is easily demonstrated as follows. Suppose L1 , L2 ∈ Kon , such
that V (L∗1 ) = ωn = V (L∗2 ), and
Ve−p (K, L1 ) = Ve−p (K, L2 ).
Define L ∈ Kon , by
1
1
e −p ? L2 ,
? L1 +
2
2
Ve−p (K, L) = Ve−p (K, L1 ) = Ve−p (K, L2 ).
L=
Since obviously,
1
1
· L∗1 +p · L∗2 ,
2
2
∗
∗
and V (L1 ) = ωn = V (L2 ), it follows from Lemma 2.1 that V (L∗ ) ≥ ωn , with equality if and only if L1 = L2 .
Thus,
p
p
Ve−p (K, L)V (L∗ ) n < Ve−p (K, L1 )V (L∗1 ) n
L∗ =
is the contradiction that would arise if it were the case that L1 6= L2 . This completes the proof.
4. Proofs of Results
In order to prove Theorem 1.11, we require the following lemma (see [5]).
Lemma 4.1. If f0 , f1 , · · · fm are (strictly) positive continuous functions defined on S n−1 and λ1 , · · · λm are
positive constants the sum of whose reciprocals is unity, then
Z
f0 (u) · · · fm (u)dS(u) ≤
S n−1
1
m Z
Y
i=1
S n−1
f0 (u)fiλi (u)dS(u)
λi
(4.1)
λm (u)
with equality if and only if there exist positive constants α1 , α2 , · · · , αm such that α1 f1λ1 (u) = · · · = αm fm
n−1
for all u ∈ S
.
Proof of Theorem 1.11. Let λi = m (1 ≤ i ≤ n),
1
f0 (u) = [e
g−p (K1 , u) . . . ge−p (Kn−m , u)] n ,
1
fi+1 (u) = [e
g−p (Kn−i , u)] n ,
by (1.6) and Lemma 4.1, we get
Z
e
G−p (K1 , . . . , Kn ) =
1
S n−1
≤
=
=
[e
g−p (K1 , u) · · · ge−p (Kn , u)] n dS(u)
m−1
Y Z
S n−1
i=0
m−1
Y Z
i=0
m−1
Y
i=0
(0 ≤ i ≤ m − 1),
S n−1
1
m
f0 (u)fi+1 (u) dS(u)
m
m
1
n
[e
g−p (K1 , u) · · · ge−p (Kn−m , u)e
g−p (Kn−i , u) ] dS(u)
e −p (K1 , . . . , Kn−m , Kn−i , . . . , Kn−i ) m1 .
G
|
{z
}
1
m
Y. Li, W. Weidong, S. Lin, J. Nonlinear Sci. Appl. 9 (2016), 1143–1152
1149
According to the equality condition of Lemma 4.1, we see that equality holds in inequality (1.12) if and
only if there exist positive constants c1 , c2 , · · · , cm such that
−p
−p
n+p
c1 ρn+p
Kn (u)ρ e (u) = c2 ρKn−1 (u)ρ e
Kn−1
Kn
−p
(u) = · · · = cm ρn+p
Kn−m+1 (u)ρ e
Kn−m+1
(u)
for all u ∈ S n−1 .
Corollary 4.2. If K1 , · · · , Kn ∈ Kon , then for p ≥ 1,
n
e −p (K1 , · · · , Kn ) ≤ G
e −p (K1 ) · · · G
e −p (Kn )
G
(4.2)
with equality if and only if there exist constants c1 , c2 , · · · , cn (not all zero) such that for all u ∈ S n−1 ,
−p
−p
n+p
c1 ρn+p
Kn (u)ρ e (u) = c2 ρKn−1 (u)ρ e
Kn−1 (u)
Kn
−p
= · · · = cn ρn+p
K1 (u)ρ e (u).
K1
Proof. Let m = n in Theorem 1.11 and by (1.1), we easily obtain Corollary 4.2.
According to the equality condition of (1.12), we see that equality holds in (4.2) if and only if there exist
constants c1 , c2 , · · · , cn (not all zero) such that for all u ∈ S n−1 ,
n+p
−p
−p
c1 ρn+p
Kn (u)ρ e (u) = c2 ρKn−1 (u)ρ e
Kn−1
Kn
Proof of Theorem 1.12. Since i < j < k, thus
k−j
k−i
k−j
−p
(u) = · · · = cn ρn+p
K1 (u)ρ e (u).
K1
> 1. From (1.6) and Hölder inequality, we obtain
j−i
e −p,i (K, L) k−i G
e −p,k (K, L) k−i
G
k−j
Z
k−i
n−i
i
1
ge−p (K, u) n ge−p (L, u) n dS(u)
=
n S n−1
j−i
Z
k−i
n−k
k
1
×
ge−p (K, u) n ge−p (L, u) n dS(u)
n S n−1
Z
k−j
(n−i)(k−j)
i(k−j) k−i
k−i
1
n(k−i)
n(k−i)
k−j
=
ge−p (L, u)
] dS(u)
[e
g−p (K, u)
n S n−1
Z
j−i
(n−k)(j−i)
k(j−i) k−i
k−i
1
×
[e
g−p (K, u) n(k−i) ge−p (L, u) n(k−i) ] j−i dS(u)
n n−1
Z S
n−j
j
1
≥
ge−p (K, u) n ge−p (L, u) n dS(u)
n S n−1
e −p,j (K, L).
=G
This gives the desired inequality (1.13). According to the equality conditions of the Hölder inequality, we
know that equality holds in (1.13) if and only if there exists a constant λ > 0 such that
ge−p (K, u)
(n−i)(k−j)
n(k−i)
ge−p (L, u)
i(k−j)
n(k−i)
k−i
k−j
k−i
(n−k)(j−i)
k(j−i)
j−i
n(k−i)
n(k−i)
= λ ge−p (K, u)
ge−p (L, u)
,
i.e. ge−p (K, u) = λe
g−p (L, u) for all u ∈ S n−1 . Thus equality holds in (1.13) if and only if K and L are dilates
of each other.
Let L = B in Theorem 1.12 and use (1.8), we obtain the following corollary.
Y. Li, W. Weidong, S. Lin, J. Nonlinear Sci. Appl. 9 (2016), 1143–1152
1150
Corollary 4.3. If K ∈ Kon , p ≥ 1, i, j, k ∈ R and i < j < k, then
e −p,j (K)k−i ≤ G
e −p,i (K)k−j G
e −p,k (K)j−i
G
(4.3)
with equality if and only if K and L are dilates of each other.
By Theorem 1.12, we also get the Minkowski inequality for the Lp -dual mixed geominimal surface area
as follows:
Corollary 4.4. If K, L ∈ Kon , n 6= p ≥ 1 and i ∈ R, then for i < 0 or i > n,
e −p,i (K, L)n ≥ G
e −p (K)n−i G
e −p (L)i ;
G
(4.4)
e −p (L)i .
e −p,i (K, L)n ≤ G
e −p (K)n−i G
G
(4.5)
for 0 < i < n,
In every case, equality holds if and only if K and L are dilates of each other.
Proof. For i < 0, take (i, j, k) = (i, 0, n) in Theorem 1.12, we have
e −p,n (K, L)−i ≥ G
e −p,0 (K, L)n−i ,
e −p,i (K, L)n G
G
(4.6)
e −p,i (K, L)n ≥ G
e −p (K)n−i G
e −p (L)i
G
(4.7)
i.e.
with equality if and only if K and L are dilates of each other.
In the same way, let (i, j, k) = (0, n, i) for i > n in Theorem 1.12, we obtain
e −p,i (K, L)n ≥ G
e −p (K)n−i G
e −p (L)i
G
(4.8)
with equality if and only if K and L are dilates of each other.
Similarly, let (i, j, k) = (0, i, n) for 0 < i < n in Theorem 1.12, we easily get
e −p,i (K, L)n ≤ G
e −p (K)n−i G
e −p (L)i
G
(4.9)
with equality if and only if K and L are dilates of each other.
Let L = B in Corollary 4.4, and notice G−p (B) = nωn by (1.9), we have that:
Corollary 4.5. If K ∈ Kon , n 6= p ≥ 1, i ∈ R, then for i < 0 or i > n,
e −p,i (K)n ≥ (nωn )i G
e −p (K)n−i ;
G
(4.10)
e −p,i (K)n ≤ (nωn )i G
e −p (K)n−i .
G
(4.11)
for 0 < i < n
In each case, equality holds if and only if K is a ball centered at the origin.
In order to prove Theorem 1.13, we require the following result (see [13]).
Lemma 4.6. If K ∈ Kcn , n ≥ p ≥ 1, then
e −p (K)G
e −p (K ∗ ) ≤ (nωn )2
G
with equality if and only if K is an ellipsoid.
Proof of Theorem 1.13. For 0 < i < n, n ≥ p ≥ 1, and by (4.5) and Lemma 4.6, we have
e −p,i (K, L)n G
e −p,i (K ∗ , L∗ )n ≤ [G
e −p (K)G
e −p (K ∗ )]n−i [G
e −p (L)G
e −p (L∗ )]i ≤ (nωn )2n
G
with equality if and only if K and L are dilated ellipsoids of each other.
Y. Li, W. Weidong, S. Lin, J. Nonlinear Sci. Appl. 9 (2016), 1143–1152
1151
For the Lp -dual geominimal surface area, Wang and Qi (see [13]) proved the following result.
Lemma 4.7. If K ∈ Kcn , p ≥ 1, then
e −p (K) ≥ n(ωn )− np V (K) n+p
n
G
(4.12)
with equality if and only if K is an ellipsoid centered at the origin.
Combining with (4.12), we can prove the following fact.
Theorem 4.8. If K, L ∈ Kcn , p ≥ 1, i ∈ R and i < 0, then
(n+p)i−pn
n2
e −p,i (K) ≥ nωn
G
V (K)
(n+p)(n−i)
n2
(4.13)
with equality if and only if K is a ball centered at the origin.
Proof. For i < 0, by (4.10) and (4.12), we get
e −p (K)n−i
e −p,i (K)n ≥ (nωn )i G
G
p
≥ (nωn )i [n(ωn )− n V (K)
(n+p)i−pn
n
= nn ωn
i.e.
(n+p)i−pn
n2
e −p,i (K) ≥ nωn
G
V (K)
V (K)
n+p
n
]n−i
(n+p)(n−i)
n
,
(n+p)(n−i)
n2
with equality if and only if K is a ball centered at the origin.
Acknowledgment
Research is supported in part by the Natural Science Foundation of China (Grant No.11371224) and
Foundation of Degree Dissertation of Master of China Three Gorges University (Grant No.2015PY070).
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