*L*_{
p
}-Dual geominimal surface area

- Wang Weidong
^{1}Email author and - Qi Chen
^{1}

**2011**:6

https://doi.org/10.1186/1029-242X-2011-6

© Weidong and Chen; licensee Springer. 2011

**Received: **1 December 2010

**Accepted: **17 June 2011

**Published: **17 June 2011

## Abstract

Lutwak proposed the notion of *L*_{
p
} -geominimal surface area according to the *L*_{
p
} -mixed volume. In this article, associated with the *L*_{
p
} -dual mixed volume, we introduce the *L*_{
p
} -dual geominimal surface area and prove some inequalities for this notion.

**2000 Mathematics Subject Classification:** 52A20 52A40.

## Keywords

*L*

_{ p }-geominimal surface area

*L*

_{ p }-mixed volume

*L*

_{ p }-dual geominimal surface area

*L*

_{ p }-dual mixed volume

## 1 Introduction and main results

Let
denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean space ℝ ^{
n
} . For the set of convex bodies containing the origin in their interiors and the set of origin-symmetric convex bodies in ℝ ^{
n
} , we write
and
, respectively. Let
denote the set of star bodies (about the origin) in *R*^{
n
} . Let *S*^{n-1}denote the unit sphere in ℝ ^{
n
} ; denote by *V* (*K*) the *n*-dimensional volume of body *K*; for the standard unit ball *B* in ℝ ^{
n
} , denote *ω*_{
n
} = *V* (*B*).

*G*(

*K*), of

*K*is defined by

Here *Q** denotes the polar of body *Q* and *V*_{1}(*M*, *N*) denotes the mixed volume of
[2].

*L*

_{ p }-mixed volume, Lutwak [3] introduced the notion of

*L*

_{ p }-geominimal surface area. For ,

*p*≥ 1, the

*L*

_{ p }-geominimal surface area,

*G*

_{ p }(

*K*), of

*K*is defined by

Here *V*_{
p
} (*M*, *N*) denotes the *L*_{
p
} -mixed volume of
[3, 4]. Obviously, if *p* = 1, *G*_{
p
} (*K*) is just the geominimal surface area *G*(*K*). Further, Lutwak [3] proved the following result for the *L*_{
p
} -geominimal surface area.

*with equality if and only if K is an ellipsoid*.

*L*

_{ p }-geominimal area ratio as follows: For , the

*L*

_{ p }-geominimal area ratio of

*K*is defined by

Lutwak [3] proved (1.3) is monotone nondecreasing in *p*, namely

*with equality if and only if K and T*_{
p
}*K are dilates*.

Here *T*_{
p
}*K* denotes the *L*_{
p
} -Petty body of
[3].

*L*

_{ p }-geominimal surface area is based on the

*L*

_{ p }-mixed volume. In this paper, associated with the

*L*

_{ p }-dual mixed volume, we give the notion of

*L*

_{ p }-dual geominimal surface area as follows: For , and

*p*≥ 1, the

*L*

_{ p }-dual geominimal surface area, , of

*K*is defined by

Here,
denotes the *L*_{
p
} -dual mixed volume of
[3].

For the *L*_{
p
} -dual geominimal surface area, we proved the following dual forms of Theorems 1.A and 1.B, respectively.

*with equality if and only if K is an ellipsoid centered at the origin*.

*with equality if and only if*
.

may be called the *L*_{
p
} -dual geominimal surface area ratio of
.

Further, we establish Blaschke-Santaló type inequality for the *L*_{
p
} -dual geominimal surface area as follows:

*with equality if and only if K is an ellipsoid*.

Finally, we give the following Brunn-Minkowski type inequality for the *L*_{
p
} -dual geominimal surface area.

*with equality if and only if K and L are dilates*.

Here *λ* ⋆ *K* + _{
-p
} *μ* ⋆ *L* denotes the *L*_{
p
} -harmonic radial combination of *K* and *L*.

The proofs of Theorems 1.1-1.3 are completed in Section 3 of this paper. In Section 4, we will give proof of Theorem 1.4.

## 2 Preliminaries

### 2.1 Support function, radial function and polar of convex bodies

where *x*·*y* denotes the standard inner product of *x* and *y*.

*K*is a compact star-shaped (about the origin) in

*R*

^{ n }, then its radial function,

*ρ*

_{ K }=

*ρ*(

*K*,·):

*R*

^{ n }\{0} → [0, ∞), is defined by [5, 6]

If *ρ*_{
K
} is continuous and positive, then *K* will be called a star body. Two star bodies *K*, *L* are said to be dilates (of one another) if *ρ*_{
K
} (*u*)*/ρ*_{
L
} (*u*) is independent of *u* ∈ *S*^{n-1}.

Here *GL*(*n*) denotes the group of general (nonsingular) linear transformations and *ϕ*^{
-τ
} denotes the reverse of transpose (transpose of reverse) of *ϕ*.

For and its polar body, the well-known Blaschke-Santaló inequality can be stated that [5]:

*with equality if and only if K is an ellipsoid*.

### 2.2 *L*_{
p
}-Mixed volume

where "·" in *ε*·*L* denotes the Firey scalar multiplication.

The *L*_{
p
} -Minkowski inequality can be stated that [4]:

*with equality for p >* 1 *if and only if K and L are dilates*, *for p* = 1 *if and only if K and L are homothetic*.

### 2.3 *L*_{
p
}-Dual mixed volume

*p*≥ 1 and

*λ*,

*μ*≥ 0 (not both zero), the

*L*

_{ p }harmonic-radial combination, of

*K*and

*L*is defined by [3]

*L*

_{ p }-harmonic radial combination of star bodies, Lutwak [3] introduced the notion of

*L*

_{ p }-dual mixed volume as follows: For ,

*p*≥ 1 and

*ε >*0, the

*L*

_{ p }-dual mixed volume, of the

*K*and

*L*is defined by [3]

*L*

_{ p }-dual mixed volume [3]:

where the integration is with respect to spherical Lebesgue measure *S* on *S*^{n- 1}.

The Minkowski's inequality for the *L*_{
p
} -dual mixed volume is that [3]

*with equality if and only if K and L are dilates*.

### 2.4 *L*_{
p
}-Curvature image

Equation (2.11) is also called Radon-Nikodym derivative, it turns out that the measure *S*_{
p
} (*K*, ·) is absolutely continuous with respect to surface area measure *S*(*K*, *·*).

*L*

_{ p }-curvature function [3]

*f*

_{ p }(

*K*, ·):

*S*

^{n-1}→ ℝ, if its

*L*

_{ p }-surface area measure

*S*

_{ p }(

*K*, ·) is absolutely continuous with respect to spherical Lebesgue measure

*S*, and

*Let*
, denote set of all bodies in
,
, respectively, that have a positive continuous curvature function.

*L*

_{ p }-curvature image as follows: For each and real

*p*≥ 1, define , the

*L*

_{ p }-curvature image of

*K*, by

Note that for *p* = 1, this definition differs from the definition of classical curvature image [3]. For the studies of classical curvature image and *L*_{
p
} -curvature image, one may see [6, 8–12].

## 3 *L*_{
p
}-Dual geominimal surface area

In this section, we research the *L*_{
p
} -dual geominimal surface area. First, we give a property of the *L*_{
p
} -dual geominimal surface area under the general linear transformation. Next, we will complete proofs of Theorems 1.1-1.3.

For the *L*_{
p
} -geominimal surface area, Lutwak [3] proved the following a property under the special linear transformation.

Here *SL*(*n*) denotes the group of special linear transformations.

Similar to Theorem 3.A, we get the following result of general linear transformation for the *L*_{
p
} -dual geominimal surface area:

Note that for *ϕ* ∈ *SL*(*n*), proof of (3.3) may be fund in [3].

This immediately yields (3.2). □

we may extend Theorem 3.A as follows:

Obviously, (3.2) is dual form of (3.4). In particular, if *ϕ* ∈ *SL*(*n*), then (3.4) is just (3.1).

Now we prove Theorems 1.1-1.3.

this yield inequality (1.5). According to the equality conditions of (2.3) and (2.10), we see that equality holds in (1.5) if and only if *K* and
are dilates and *Q* is an ellipsoid, i.e. *K* is an ellipsoid centered at the origin. □

Compare to inequalities (1.2) and (1.5), we easily get that

*with equality if and only if K is an ellipsoid centered at the origin*.

According to equality condition in the Hölder inequality, we know that equality holds in (3.5) if and only if *K* and *Q* are dilates.

This gives inequality (1.6).

Because of in inequality (3.6), this together with equality condition of (3.5), we see that equality holds in (1.6) if and only if . □

According to the equality condition of (2.3), we see that equality holds in (1.7) if and only if *K* is an ellipsoid. □

Associated with the *L*_{
p
} -curvature image of convex bodies, we may give a result more better than inequality (1.5) of Theorem 1.1.

*with equality if and only if*
.

This yields (3.9). According to the equality condition in inequality (2.4), we see that equality holds in inequality (3.9) if and only if *K* and *Q** are dilates. Since
, equality holds in inequality (3.9) if and only if
. □

*with equality if and only if K is an ellipsoid*.

*with equality if and only if K is an ellipsoid*.

Inequality (3.12) just is inequality (1.5) for the *L*_{
p
} -curvature image.

In addition, by (1.2) and (3.9), we also have that

*with equality if and only if K is an ellipsoid*.

## 4 Brunn-Minkowski type inequalities

In this section, we first prove Theorem 1.4. Next, associated with the *L*_{
p
} -harmonic radial combination of star bodies, we give another Brunn-Minkowski type inequality for the *L*_{
p
} -dual geominimal surface area.

*with equality if and only if K and L are dilates*.

*Proof*. Since

*-*(

*n*+

*p*)/

*p <*0, thus by (2.5), (2.8) and Minkowski's integral inequality (see [14]), we have for any ,

According to the equality condition of Minkowski's integral inequality, we see that equality holds in (4.1) if and only if *K* and *L* are dilates. □

This yields inequality (1.8).

By the equality condition of (4.1) we know that equality holds in (1.8) if and only if *K* and *L* are dilates. □

*L*

_{ p }-radial combination can be introduced as follows: For ,

*p*≥ 1 and

*λ*,

*μ*≥ 0 (not both zero), the

*L*

_{ p }-radial combination, , of

*K*and

*L*is defined by [15]

Under the definition (4.2) of *L*_{
p
} -radial combination, we also obtain the following Brunn-Minkowski type inequality for the *L*_{
p
} -dual geominimal surface area.

*with equality if and only if K and L are dilates*.

The equality holds if and only if
are dilates with *K* and *L*, respectively. This mean that equality holds in (4.3) if and only if *K* and *L* are dilates. □

## Declarations

### Acknowledgements

We wish to thank the referees for this paper. Research is supported in part by the Natural Science Foundation of China (Grant No. 10671117) and Science Foundation of China Three Gorges University.

## Authors’ Affiliations

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