The following Lemmas will be required to prove our main theorems.
Lemma 1
[5]If K(f ) is the Aleksandrov body associated with f ∈ C+(Sn- 1), then hK(f)= f almost everywhere with respect to the measure S(K(f ), ·) on Sn- 1.
Obviously, if K(f ) is the Aleksandrov body corresponding to a given function f ∈ C+(Sn- 1), its support function has the property that 0 < h
K
≤ f and V (f ) = V (hK(f )).
Lemma 2
[5]If p ≥ 1, K(f ) is the Aleksandrov body associated with f ∈ C+(Sn- 1), then V (f ) = V (K(f )) = V
p
(K(f ), f ), i.e.
Lemma 3
[5]If, f ∈ C+(Sn- 1), then, for p ≥ 1,
(3.1)
We get the following Brunn-Minkowski inequality for the Aleksandrov bodies associated with positive continuous functions.
Lemma 4
If f, g ∈ C+(Sn- 1), and λ, μ ∈ℝ+, then
(3.2)
with equality if and only if there exist a constant c > 0 and t ≥ 0, such that f = cg + t, almost everywhere with respect to S(K(f ), ·) on Sn- 1.
Proof
Since f, g ∈ C+(Sn- 1), from (2.11), (2.12) and the Brunn-Minkowski inequality (see [21]), we get
(3.3)
The equality condition in (3.3) is that f, g are the support functions of K(f ) and K(g) which are homothetic, respectively.
From Lemma 1 and Lemma 2, we get the following result
(3.4)
with equality if and only if there exist a constant c > 0 and t ≥ 0, such that f = cg + t, almost everywhere with respect to S(K(f ), ·) on Sn- 1.
An immediate consequence of the definition of a Firey linear combination, and the integral representation (2.13), is that for , the p- mixed volume
is Firey linear.
Lemma 5
If p ≥ 1, , f, g ∈ C+(Sn- 1), and λ, μ ∈ ℝ+, then
(3.5)
Proof
From (2.13), (2.14), we obtain
In the following, we will prove the p- Minkowski inequality for the Aleksandrov bodies associated with positive continuous functions.
Proof of Theorem 1.
Firstly, let p = 1 in Lemma 3, we get
let , we have
Let
we see that
From Lemma 4, we know that f is concave, i.e.
Thus,
(3.6)
According to the equality condition in inequality (3.3), and using Lemma 1 and Lemma 2, we have the equality holds in inequality (3.6), if and only if there exist a constant c > 0 and t ≥ 0, such that h
Q
= cf + t, almost everywhere with respect to S(Q, ·) on Sn- 1.
Secondly, from the Hölder inequality (see [22]), together with the integral representations (2.13) and (2.6), we obtain
when this combined with inequality (3.6), we have
(3.7)
To obtain the equality conditions, we note that there is equality in Hölder's inequality precisely when V1(Q, f )h
Q
= V (Q)f, almost everywhere with respect to the measure S(Q, ·) on Sn- 1. Combining the equality conditions in (3.6), and using Lemma 1, it shows that the equality holds if and only if there exists a constant c > 0 such that h
Q
= cf, almost everywhere with respect to S(Q, ·) on Sn- 1.
Using the above Lemmas and Theorem 1, we can get the following Corollaries describing the uniqueness results.
Corollary 1
Suppose K, , and ⊂ C+ (Sn- 1) is a class of functions such that h
K
, h
L
∈ . (i) If n ≠ p > 1, and V
p
(K, f ) = V
p
(L, f ), for all f ∈ , then K = L. (ii) If p = n, and V
p
(K, f ) ≥ V
p
(L, f ), for all f ∈ , then K and L are dilates, and hence
Proof
If n ≠ p > 1, take f = h
K
, and from (2.13), Lemma 2 and Theorem 1, we get
Hence,
Similarly, take f = h
L
, we get
In view of the equality conditions of Theorem 1, we obtain that K = L.
If n = p, the hypothesis together with Theorem 1, we have
with equality in the right inequality implying that L and K(f ) are dilates. Take f = h
K
, since n = p, the terms on the left and right are identical, and thus, K and L must dilates; hence,
Corollary 2
Suppose f, g ∈ C+(Sn- 1), and ⊂ C+ (Sn- 1) is a class of functions such that f, g ∈ . If p > 1, and
then f = g almost everywhere on Sn- 1.
Proof
Since f, g ∈ C+(Sn- 1), according to (2.11), we denote two Aleksandrov bodies K(f ) and K(g). From the hypothesis, taking Q = K(f ), and using Lemma 2 and Theorem 1, we get
then,
Similarly, take Q = K(g), we get
From the equality conditions of Theorem 1, we obtain
In view of the definition of Aleksandrov body, and using Lemma 1, then
Corollary 3
Suppose n ≠ p > 1, and f, g ∈ C+(Sn- 1), such that S
p
(K(f ), ·) ≤ S
p
(K(g), ·).
-
(i)
If V (f ) ≥ V (g), and p < n, then f = g almost everywhere on Sn- 1.
-
(ii)
If V (f ) ≤ V (g), and p > n, then f = g almost everywhere on S n- 1.
Proof
Suppose a function φ ∈ C+(Sn- 1), and n ≠ p > 1, since S
p
(K(f ), ·) ≤ S
p
(K(g), ·), it follows from the integral representation (2.13) and (2.8) that
As before, take φ = hK(g), from Lemma 1, Lemma 2, and Theorem 1, we get
Applying the hypothesis, and from the definition of the Aleksandrov body and Lemma 1, we obtain the desired results.
Corollary 4
Suppose n ≠ p ≥ 1, f, g ∈ C+(Sn- 1), and ⊂ C+ (Sn- 1) is a class of functions such that f, g ∈ . If
then f = g almost everywhere on Sn- 1.
Proof
According to (2.11), we denote two Aleksandrov bodies K(f ) and K(g). From the hypothesis, taking K = K(f ) and K = K(g), and combining with Lemma 2 and Theorem 1, respectively, we obtain
Hence, in view of the equality conditions of Theorem 1, the definition of Aleksandrov body, and Lemma 1, we get the desired result
Now, the p- Brunn-Minkowski inequality for the p- Aleksandrov bodies and the Aleksandrov bodies associated with positive continuous functions is established as following.
Proof of Theorem 2.
From Lemma 5 and Theorem 1, we get
with equality if and only if K(f) and K(g) are dilates of Q.
Now, take Q = K
p
(λ · f +
p
μ · g), use (2.10), and recall V (f ) = V (K(f )) = V
p
(K(f ), f ), we have
Also, we note that the equality holds, if and only if K(f ) and K(g) are dilates. Using Lemma 1, we get the condition of equality holds if and only if there exists a constant c > 0 such that f = cg, almost everywhere with respect to S(K(f ), ·) on Sn- 1.
Then, we will prove Theorem 3 by using the generalized Blaschke linear combination.
Proof of Theorem 3.
Suppose a function φ ∈ C+(Sn- 1), and n ≠ p ≥ 1, from the integral representation (2.13), (2.8), and (2.4), it follows that for K(f ), ,
(3.8)
which together with Theorem 1, yields
(3.9)
with equality if and only if K(f); K(g) and K(φ ) are dilates.
Now, take , recall Vp(K, h
K
) = V (K ), and from Lemma 2, we get
In view of (2.16), we have
Hence, we get
From Lemma 2 again, we obtain
In view of the equality condition (3.9), and from Lemma 1, we get the equality holds if and only if there exists a constant c > 0 such that f = cg, almost everywhere with respect to S(K(f ), ·) on Sn- 1.
Remark 1
The case p = 1 of the inequality of Theorem 3 is
(3.10)
with equality if and only if there exists a constant c > 0 such that f = cg, almost everywhere with respect to S(K(f ), ·) on Sn- 1.
The above inequality (3.10) is just the Kneser-Süss inequality type for the Aleksandrov bodies associated with positive continuous functions.
Actually, from these above proofs, we see Brunn-Minkowski inequality, Minkowski inequality, and Knesser-Süss inequality are equivalent.