Abstract
In the paper, we first give an improvement of Minkowski integral inequality. As an application, we get new Brunn-Minkowski-type inequalities for dual mixed volumes.
2000 Mathematics Subject Classification: 52A30, 52A40, 26D15
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1 Improvement of Minkowski's inequality
The well-known inequality due to Minkowski can be stated as follows ([1], pp. 19-20, [2], p. 31]):
Theorem 1.1 Let f(x), g(x) ≥ 0 and p > 1, then
with equality if and only if f and g are proportional, and if p < 1 (p ≠ 0), then
with equality if and only if f and g are proportional. For p < 0, we assume that f(x), g(x) > 0.
An (almost) improvement of Minkowski's inequality, for p ∈ ℝ\{0}, is obtained in the following Theorem:
Theorem 1.2 Let f(x), g(x) ≥ 0 and p > 0, or f(x), g(x) > 0 and p < 0. Let s, t ∈ ℝ\{0}, and s ≠ t. Then
-
(i)
Let p, s, t ∈ ℝ be different, such that s, t > 1 and (s - t)/(p - t) > 1. Then
(1.3)
with equality if and only if f(x) and g(x) are constant, or 1/p = (1/s + 1/t)/2 and f(x) and g(x) are proportional.
-
(ii)
Let p, s, t ∈ ℝ be different, such that s, t < 1 and s, t ≠ 0, and (s - t)/(p - t) < 1. Then
(1.4)
with equality if and only if f(x) and g(x) are constant, or 1/p = (1/s + 1/t)/2 and f(x) and g(x) are proportional.
Proof (i) We have (s - t)/(p - t) > 1, and in view of
By using Hölder's inequality (see [1] or [2]) with indices (s - t)/(p - t) and (s - t)/(s - p), we have
with equality if and only if (f(x) + g(x))s(p - t)/(s - t)and (f(x) + g(x))t(s - p)/(s - t)are proportional, i.e., either f(x) + g(x) is constant or the exponents are equal, i.e., 1/p = (1/s + 1/t)/2.
On the other hand, by using Minkowski's inequality for s > 1 and t > 1, respectively, we obtain
with equality if and only if f(x) and g(x) are proportional, and
with equality if and only if f(x) and g(x) are proportional.
From (1.5), (1.6) and (1.7), (1.3) easily follows. From the equality conditions of (1.5), (1.6) and (1.7), the case of equality stated in (i) follows.
-
(ii)
We have (s - t)/(p - t) < 1. Similar to the above proof, we have
(1.8)
with equality if and only if either f(x) + g(x) is constant or 1/p = (1/s + 1/t)/2.
On the other hand, in view of Minkowski's inequality for the cases of 0 < s < 1 and 0 < t < 1,
with equality if and only if f(x) and g(x) are proportional, and
with equality if and only if f(x) and g(x) are proportional.
The inequality (1.4) easily follows, with equality as stated in (ii). ■
Remark 1.3 For (i) of Theorem 1.2, for p > 1, letting s = p + ε, t = p - ε, when p, s, t are different, s, t > 1, and (s - t)/(p - t) / 2 > 1, and letting ε → 0, we get (1.1).
For (ii) of Theorem 1.2, for p < 1 and p ≠ 0, s = p + ε, t = p + 2ε, when p, s, t are different, s, t < 1 and s, t ≠ 0, and (s - t)/(p - t) = 1/2 < 1, and letting ε → 0, we get (1.2).
2 An application
The setting for this paper is n-dimensional Euclidean space ℝn(n > 2). Associated with a compact subset K of ℝn, which is star-shaped with respect to the origin, is its radial function ρ(K, ·): Sn - 1→ ℝ, defined for u ∈ Sn - 1, by
If ρ(K, ·) is positive and continuous, K will be called a star body. Let denote the set of star bodies in ℝn. Let denote the radial Hausdorff metric, that is defined as follows: if , then (see e.g. [3]).
If and λ1, ...,λ r ∈ ℝ, then the radial Minkowski linear combination, , is defined by Lutwak (see [4]), as . Here, equals λ1x1 + ⋯ +λ r x r if x 1, ⋯, x r belong to a linear 1-subspace of ℝn, and is 0 else. It has the following important property, for and λ, μ ≥ 0
For and λ1, ..., λ r ≥ 0, the volume of the radial Minkowski linear combination is a homogeneous n th-degree polynomial in the λ i ,
where the sum is taken over all n-tuples (i1, ..., i n ) whose entries are positive integers not exceeding r. If we require the coefficients of the polynomial in (2.2) to be symmetric in their argument, then they are uniquely determined. The coefficient is positive and depends only on the star bodies . It is written as and is called the dual mixed volume of . If K1 = ⋯ = K n - i = K, Kn - i+1= ⋯ = K n = L, the dual mixed volumes are written as . In particular, for B the unit ball about o, is written as (see [5]).
For , the dual mixed volumes were given by Lutwak (see [6]), as
For and i ∈ ℝ, the i th dual mixed volume of K and L, , is defined by,
From (2.4), taking in consideration ρ(B, u) = 1, if , and i ∈ ℝ
The well-known Brunn-Minkowski-type inequality for dual mixed volumes can be stated as follows [6]:
Theorem 2.1 Let , and i < n - 1. Then,
with equality if and only if K and L are dilates.
The inequality is reversed for i > n - 1 and i ≠ n.
In the following, we establish new Brunn-Minkowski-type inequalities for dual mixed volumes.
Theorem 2.2 Let and i, j, k ∈ ℝ.
-
(i)
Let i, j, k ∈ ℝ be different, such that j, k < n - 1, and (j - k)/(i - k) > 1. Then
(2.7)
with equality if and only if K and L are balls, or 1/(n - i) = [1/(n - j) + 1/(n - k)]/2, and K and L are dilates.
-
(ii)
Let i, j, k ∈ ℝ be different, such that j, k > n - 1 and j, k ≠ n, and (j - k)/(i - k) < 1. Then
(2.8)
with equality if and only if K and L are balls, or 1/(n - i) = [1/(n - j) + 1/(n - k)]/2, and K and L are dilates..
Proof We begin with the proof of (i). From (2.1), (2.5) and (1.3), we have
with equality if and only if as stated in (i).
Similarly, case (ii) of Theorem 2.2 easily follows. ■
Remark 2.3 For (i) of Theorem 2.2, for n - i > 1, letting s = n - i + ε, t = n - i - ε, when i, j, k are different, n - j, n - k > 1, and (k - j)/(k - i) = 2 > 1, and letting ε → 0, we get the following result: Let , and i < n - 1. Then,
with equality if and only if K and L are dilates.
This is just the well-known inequality (2.6) in Theorem 2.1.
For (ii) of Theorem 2.2, for n - i < 1 and n - i ≠ 0, s = n - i + ε, t = n - i + 2ε, when i, j, k are different, n - j, n - k < 1 and n - j, n - k ≠ 0, and (k - j)/(k - i) = 1/2 < 1, and letting ε → 0, we get the following result:
Let , and i < n - 1 and i ≠ n. Then,
with equality if and only if K and L are dilates.
This is just an reversed form of inequality (2.6).
References
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Acknowledgements
The authors wish to thank the referee for his many excellent suggestions for improving the original manuscript.
This Research is supported by National Natural Science Foundation of China(10971205) and in part by a HKU URC grant.
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Authors' contributions
C-JZ and W-SC jointly contributed to the main results Theorems 1.2 and 2.2, Both authors read and approved the final manuscript.
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Zhao, CJ., Cheung, WS. On Minkowski's inequality and its application. J Inequal Appl 2011, 71 (2011). https://doi.org/10.1186/1029-242X-2011-71
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DOI: https://doi.org/10.1186/1029-242X-2011-71