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- JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2015-0026
Mathematical and Physical Sci., 2015, Vol. 60, No. 7, pp. 15-20
This paper is available online at http://stdb.hnue.edu.vn
ADDITION SOME INEQUALITY OF VOLUMES MIXED IN GEOMETRY
Lai Duc Nam
Yen Bai Teacher’s Training College, Yen Bai
Abstract. The issue norms for all volume preserving affine transformations for optimal
Sobolev is always been interested in research. In this paper, we present some results of
inequalities of volume mixed in geometry.
Keywords: Lp Sobolev inequality, geometry inequalities.
1. Introduction
The classical sharp Lp Sobolev inequality states that if f ∈ W 1,p (Rn ), with
real p satisfying 1 ≤ p < n, then
k∇f kp ≥ αn,p kf k np (1.1)
n−p
where k · kq denotes the usual Lq norm for functions on Rn . The optimal constants αn,p in this
inequality are due to Federer and Fleming [1] for p = 1 and to Aubin [2] for 1 < p < n. For
strengthened versions of (1.1), see, e.g. [3, 4, 5] and the references therein.
Recently, Zhang [6] (for p = 1) and Lutwak, Yang, and Zhang [7] (for 1 < p < n)
formulated and proved a sharp affine Lp Sobolev inequality. This remarkable inequality is invariant
under all affine transformations of Rn , while the classical Lp Sobolev inequality (1.1) is invariant
only under rigid motions.
In the affine Lp Sobolev inequality the Lp norm of the Euclidean length of the gradient is
replaced by an affine invariant of functions, the Lp affine energy, defined, for f ∈ W 1,p (Rn ), by
Z −1/n
Ep (f ) = cn,p kDu f k−n
p du (1.2)
S n−1
nκ κ
where cn,p = (nκn )1/n ( 2κn+p−2n p−1 1/p
) with κn = π n/2 /Γ(1 + n2 ) and Du f is the directional
derivative of f in the direction u.
The sharp affine Lp Sobolev inequality of Zhang [6] and Lutwak, Yang, and Zhang [7]
states that if f ∈ W 1,p (Rn ), 1 ≤ p < n, then
Ep (f ) ≥ αn,p kf k np
n−p
Received October 10, 2015. Accepted November 30, 2015.
Contact Lai Duc Nam, e-mail address: nam.laiduc@gmail.com
15
- Lai Duc Nam
It is shown in [7;33] that
k∇f kp ≥ Ep (f ). (1.3)
Hence the sharp affine Lp Sobolev inequality (1.2) is stronger than the classical affine Lp Sobolev
inequality (1.1). We emphasize the remarkable and important fact that Ep (f ) is invariant under
volume preserving affine transformations on Rn . In contrast, k∇f kp is invariant only under rigid
motions.
fp ∈ SL(n) such that
We also show a reverse inequality of (1.3), there exist a T
fp )kp ≤ Ep (f ),
cp k∇(f ◦ T
where cp is a constant only depending on p.
The technique we adopt is the solution for the function Minkowski problem
on W 1,p (Rn ) established by Lutwak, Yang and Zhang [8]. This technique plays a fundamental
role in the newly emerged affine Sobolev inequalities (see, e.g., [6, 8, 9]). The importance of this
technique is that it convert analytic inequalities to geometry inequalities. Note that the geometry
behind the sharp Lp Sobolev inequality is the isoperimetric inequality, while the affine Lp Sobolev
inequality is equivalent to the Lp Petty projection inequality established in [10, 11].
2. Notations and preliminaries
We work in Rn , which is equipped with a Euclidean structure x · y for x, y ∈ Rn . We
write B2n for the Euclidean unit ball and S n−1 for the unit sphere.
A convex body is a compact convex set in Rn which is throughout assumed to contain the
origin in its interior. We denote by Kon the space of convex bodies equipped with the Hausdorff
metric. Each convex body K is uniquely determined by its support function hK = h(K, ·) : Rn →
R defined by
hK (x) = h(K, x) := max{x · y : y ∈ K}.
Let k · kK : Rn → [0, ∞) denote the Minkowski functional of K ∈ Kon ; i.e., kxkK = min{λ ≥
0 : x ∈ λK}.
The polar set K ∗ of K ∈ Kon is the convex body defined by
K ∗ = {x ∈ Rn : x · y ≤ 1 for all y ∈ K}.
If K ∈ Kon , then it follows from the definitions of support functions and Minkowski functionals,
and the definition of polar body, that
h(K ∗ , ·) = k · kK .
For p ≥ 1, K, L ∈ Kon , the Lp Minkowski combination K +p L is the convex body defined
by
h(K +p L, ·)p = h(K, ·)p + h(L, ·)p .
Introduced by Firey in the 1960’s, this notion is the basis of what has become known as
the Lp Brunn-Minkowski theory (or the Brunn-Minkowski-Firey theory).
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- Addition some inequality of volumes mixed in geometry
The Lp mixed volume Vp (K, L) of K, L ∈ Kon is defined in [12] by
1
p V (K +p ε p L) − V (K)
Vp (K, L) = lim .
n ε→0+ ε
In particular,
Vp (K, K) = V (K)
for every convex body K.
It was shown in [12] that corresponding to each convex body K ∈ Kon , there is a positive
Borel measure on S n−1 , the Lp surface area measure Sp (K, ·) of K, such that for every L ∈ Kon ,
Z
1
Vp (K, L) = hpL (u)dSp (K, u).
n S n−1
The measure S1 (K, ·) is just the surface area measure of K. Recall that for a Borel set ω ⊂ S n−1 ,
the surface area measure S(K, ω) is the (n − 1)-dimensional Hausdorff measure of the set of
all boundary points of K for which there exists a normal vector of K belonging to ω. Moreover,
the Lp surface area measure is absolutely continuous with respect to S(K, ·):
dSp (K, u) = hK (u)1−p dS(K, u) u ∈ S n−1 .
Note that
Sp (tK, ·) = tn−p Sp (K, ·) (2.1)
for all t > 0 and convex bodies K.
3. Proof of main results
Following the definition of Lutwak, Yang and Zhang [8], we define the
e p K of K by
normalized Lp projection body Π
Z
e p K, u)p = 1
h(Π |u · v|p dSp (K, v). (3.1)
V (K) S n−1
Theorem 3.1. Let K ∈ Kon and p ≥ 1, then
n
e ∗p K) ≥ κn nκp−1 κn V (K) p .
V (Π
2κn+p−2 Sp (K)
Proof. Using (3.1), the polar coordinate formula for volume, H¨older inequality and Fubini
17
- Lai Duc Nam
theorem, we obtain
Z
e ∗p K) = 1 e p K, u)−n du
V (Π h(Π
n S n−1
1 Z − n
1+ n
≥ κn p h(Πe p K, u)p du p
n S n−1
Z Z − n
1+ n 1
|u · v|p dSp (K, v)du
p
= κn p
nV (K) S n−1 S n−1
1 Z Z − n
1+ n p
= κn p
|u · v|p dudSp (K, v)
nV (K) S n−1 S n−1
2κ Z − n
1+ n
p n+p−2 p
= κn dSp (K, v)
nκp−1 V (K) S n−1
nκ κ V (K) n
p−1 n p
= κn
2κn+p−2 Sp (K)
Theorem 3.2. Let K ∈ Kon and p ≥ 1. If Sp (K, ·) is isotropic, then
S (K) 1 1
p p
e p K, u) ≤ √1 Sp (K) p ,
≤ h(Π 1 ≤ p ≤ 2,
nV (K) n nV (K)
and
S (K) 1 1
p p
e p K, u) ≥ √1 Sp (K) p ,
≥ h(Π p ≥ 2,
nV (K) n nV (K)
for every u ∈ S n−1 .
Proof. For the case that p = 2, by the definition of the normalized Lp projection body (3.1), we
have p
h(Πe 2 K, u) = S2 (K)/nV (K).
For 1 ≤ p ≤ 2, by (3.1) and the H¨older inequality, we obtain
1 Z 1
h(Πe p K, u) = |u · v|p dSp (K, v)
p
V (K) S n−1
h 1 Z p i1
p p
2
≤ |u · v|2 dSp (K, v) Sp (K)1− 2
V (K) S n−1
1 Sp (K) p1
= √
n nV (K)
for every u ∈ S n−1 . On the other hand, we have
1 Z 1 S (K) 1
e p K, u) ≥ p p p
h(Π |u · v|2 dSp (K, v) =
V (K) S n−1 nV (K)
for every u ∈ S n−1 . Similarly, we can get the result for p ≥ 2.
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- Addition some inequality of volumes mixed in geometry
In 2006, Lutwak, Yang and Zhang [8] proposed and proved the following functional
Minkowski problem on W 1,p (Rn ).
Theorem 3.3. Given 1 ≤ p < ∞ and a function f ∈ W 1,p (Rn ), there exists a unique
origin-symmetric convex body hf ip that
Z Z
1
Φ(−∇f (x))p dx = Φ(v)p dSp (hf ip , v), (3.2)
R n V (hf ip ) S n−1
for every even continuous function Φ : Rn → [0, ∞) that is homogeneous of degree 1.
Like the known result that the affine Lp Sobolev inequality implies the Lp Petty projection
inequality, we will show that our main theorems implies corresponding geometry theorems listed
in [13]. Similar to the proof of [14, Lemma 3], we show the following theorem.
Theorem 3.4. If K ∈ Kon and f = g(kxkK ) with g ∈ C 1 (0, ∞), then for p ≥ 1, hf ip is a dilate
of K, that is
−1 e
hf ip = c(f ) p K
R∞
e = K/|K|.
where c(f ) = 0 tn−1 |g′ (t)|p dt and K
Proof. Since hK ∗ is Lipschitz (and therefore differentiable almost everywhere) and hK ∗ (x) =
1 on ∂K, then for almost every x ∈ ∂K,
∇hK ∗ (x)
νK (x) = ,
|∇hK ∗ (x)|
where νK (x) is the outer unit normal vector of K at the point x.
Note that hK (∇hK ∗ (x)) = 1, for almost every x ∈ Rn . Hence we have
1
hK (νK (x)) = . (3.3)
|∇hK o (x)|
Then by (3.2), the co-area formula (see, e.g., [15;258]) applied to hK ∗ (·), the fact that ∇hKe ∗ is
homogeneous of degree 0 and (3.3), we get
Z
1
Φp (u)dSp (hf ip , u)
V (hf ip ) S n−1
Z
= Φp (∇f (x))dx
n
ZR
= Φp (g ′ (hK ∗ (x))∇hK ∗ (x)dx
n
ZR∞ Z ∇h ∗ (x)
K
= tn−1 |g ′ (t)|p Φp |∇hK o (x)|p−1 dHn−1 (x)dt
0 ∂K |∇hK ∗ (x)|
Z ∞ Z
n−1 ′ p
= t |g (t)| dt Φp (νK (x))hK (νK (x))1−p dHn−1 (x)
0 ∂K
Z ∞ Z
= tn−1 |g ′ (t)|p dt Φp (u)dSp (K, u)
0 S n−1
19
- Lai Duc Nam
for every even continuous function Φ that is homogeneous of degree 1. Thus, the uniqueness of
the solution of the even Lp Minkowski problem [12] and (2.1) imply that
hf ip 1
1 = c(f ) n−p K,
V (hf ip ) n−p
and hence 1
hf ip = c(f )− p K
e
R∞
where c(f ) = e = K/V (K)1/p .
tn−1 |g′ (t)|p dt and K
0
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