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 Rn . Let Sn-1denote 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).

The notion of geominimal surface area was given by Petty [1]. For , the geominimal surface area, G(K), of K is defined by

Here Q* denotes the polar of body Q and V1(M, N) denotes the mixed volume of [2].

According to the 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

(1.1)

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.

Theorem 1.A. If, p ≥ 1, then

(1.2)

with equality if and only if K is an ellipsoid.

Lutwak [3] also defined the L p -geominimal area ratio as follows: For , the L p -geominimal area ratio of K is defined by

(1.3)

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

Theorem 1.B. If, 1 ≤ p < q, then

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].

Above, the definition of 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

(1.4)

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.

Theorem 1.1. If, p ≥ 1, then

(1.5)

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

Theorem 1.2. If, 1 ≤ p < q, then

(1.6)

with equality if and only if.

Here

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:

Theorem 1.3. If, n ≥ p ≥ 1, then

(1.7)

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.

Theorem 1.4. If, p ≥ 1 and λ, μ ≥ 0 (not both zero), then

(1.8)

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

If , then its support function, h K = h(K,·): ℝ n → (-∞, ∞), is defined by [5, 6]

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

If K is a compact star-shaped (about the origin) in Rn , then its radial function, ρ K = ρ (K,·): Rn \{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 ∈ Sn-1.

If , the polar body, K*, of K is defined by [5, 6]

(2.1)

For , if ϕ ∈ GL(n), then by (2.1) we know that

(2.2)

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]:

Theorem 2.A. If, then

(2.3)

with equality if and only if K is an ellipsoid.

2.2 L p -Mixed volume

For and ε > 0, the Firey L p -combination is defined by [7]

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

If , then for p ≥ 1, the L p -mixed volume, V p (K, L), of K and L is defined by [4]

The L p -Minkowski inequality can be stated that [4]:

Theorem 2.B. Ifand p ≥ 1 then

(2.4)

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

For , p ≥ 1 and λ, μ ≥ 0 (not both zero), the L p harmonic-radial combination, of K and L is defined by [3]

(2.5)

From (2.5), for ϕ ∈ GL(n), we have that

(2.6)

Associated with the 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]

(2.7)

The definition above and Hospital's role give the following integral representation of the L p -dual mixed volume [3]:

(2.8)

where the integration is with respect to spherical Lebesgue measure S on Sn- 1.

From the formula (2.8), we get

(2.9)

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

Theorem 2.C. Let, p ≥ 1, then

(2.10)

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

2.4 L p -Curvature image

For , and real p ≥ 1, the L p -surface area measure, S p (K, ·), of K is defined by [4]

(2.11)

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, ·).

A convex body is said to have an L p -curvature function [3]f p (K, ·): Sn-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.

Lutwak [3] showed the notion of 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.

Theorem 3.A. For, p ≥ 1, if ϕ ∈ SL(n), then

(3.1)

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:

Theorem 3.1. For, p ≥ 1, if ϕ ∈ GL(n), then

(3.2)

Lemma 3.1. Ifand p ≥ 1, then for ϕ ∈ GL(n),

(3.3)

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

Proof. From (2.6), (2.7) and notice the fact V (ϕ K) = |detϕ|V (K), we have

□

Proof of Theorem 3.1. From (1.4), (3.3) and (2.2), we have

This immediately yields (3.2). □

Actually, using definition (1.1) and fact [13]: Ifand p ≥ 1, then for ϕ ∈ GL(n),

we may extend Theorem 3.A as follows:

Theorem 3.2. For, p ≥ 1, if ϕ ∈ GL(n), then

(3.4)

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.

Proof of Theorem 1.1. From (2.10) and Blaschke-Santaló inequality (2.3), we have that

Hence, using definition (1.4), we know

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

Corollary 3.1. For, p ≥ 1, then for n > p,

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

Proof of Theorem 1.2. Using the Hölder inequality, (2.8) and (2.9), we obtain

that is

(3.5)

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.

From definition (1.4) of , we obtain

(3.6)

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 . □

Proof of Theorem 1.3. From definition (1.4), it follows that for ,

Since , taking K for Q, and using (2.9), we can get

(3.7)

Similarly,

(3.8)

From (3.7) and (3.8), we get

Hence, for n ≥ p using (2.3), we obtain

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.

Theorem 3.3. If, p ≥ 1, then

(3.9)

with equality if and only if.

Lemma 3.2[3]. If, p ≥ 1, then for any,

(3.10)

Proof of Theorem 3.3. From (1.4), (3.10) and (2.4), we have that

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 . □

Recall that Lutwak [3] proved that ifand p ≥ 1, then

(3.11)

with equality if and only if K is an ellipsoid.

From (3.9) and (3.11), we easily get that ifand p ≥ 1, then

(3.12)

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

Corollary 3.2. If, p ≥ 1, then

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.

Lemma 4.1. If, p ≥ 1 and λ, μ ≥ 0 (not both zero) then for any,

(4.1)

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. □

Proof of Theorem 1.4. From definition (1.4) and inequality (4.1), we obtain

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. □

The notion of 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]

(4.2)

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.

Theorem 4.1. If, p ≥ 1 and λ, μ ≥ 0 (not both zero), then

(4.3)

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

Proof. From definitions (1.4), (4.2) and formula (2.8), we have

Thus

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. □