Abstract
Following the discovery of general \(L_{p}\)-projection bodies by Ludwig, Haberl and Schuster determined the extremum of the volume of the polars of this family of \(L_{p}\)-projection bodies. In this paper, the result of Haberl and Schuster is extended to all dual quermassintegrals, and a dual counterpart for the quermassintegrals of general \(L_{p}\)-projection bodies is also obtained. Moreover, the extremum of the \(L_{q}\)-dual affine surface areas of polars of general \(L_{p}\)-projection bodies are determined.
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1 Introduction
Let \(\mathcal{K}^{n}\) denote the set of convex bodies (compact, convex subsets with nonempty interiors) in the n-dimensional Euclidean space \(\mathbb{R}^{n}\). For the set of convex bodies containing the origin in their interiors and the set of origin-symmetric convex bodies in \(\mathbb{R}^{n}\), we write \({\mathcal {K}}^{n}_{\mathrm {o}}\) and \({\mathcal {K}}_{\mathrm{os}}^{n}\), respectively. Let \(S^{n-1}\) denote the unit sphere in \(\mathbb{R}^{n}\) and denote by \(V(K)\) the n-dimensional volume of the body K. For the standard unit ball B in \(\mathbb{R}^{n}\), write \(V(B)=\omega_{n}\).
For \(K\in{\mathcal {K}}^{n}\), its support function \(h_{K}=h(K,\cdot): \mathbb{R}^{n} \longrightarrow (-\infty,+\infty)\) is defined by (see [1])
where \(x\cdot y\) denotes the standard inner product of x and y.
The projection body of a convex body was introduced by Minkowski at the turn of the previous century. For \(K\in{\mathcal {K}}^{n}\), the projection body ΠK of K is the origin-symmetric convex body whose support function is defined by (see [1])
for all \(u\in{S}^{n-1}\). Here, \(S(K, \cdot)\) denotes the surface area measure of the convex body K. Classical projection bodies are a very important notion in the Brunn-Minkowski theory. During the past four decades, a number of important results regarding classical projection bodies were obtained (see [1–12]).
The notion of an \(L_{p}\)-projection body was introduced by Lutwak, Yang, and Zhang [13]. For \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\) and \(p\geq1\), the \(L_{p}\)-projection body \(\Pi_{p}K\) of K is the origin-symmetric convex body whose support function is given by
for all \(u\in{S}^{n-1}\). Here
with \(c_{n,p}=\omega_{n+p}/\omega_{2}\omega_{n}\omega_{p-1}\), and \(S_{p}(K,\cdot)\) is the \(L_{p}\)-surface area measure of K that has the Radon-Nikodym derivative
The unusual normalization of definition (1.1) is chosen so that for the unit ball B, we have \(\Pi_{p}B = B\). In particular, for \(p=1\), \(\Pi_{1}K\) is just the classical projection body ΠK of K under the different normalization of definition (1.1).
\(L_{p}\)-projection bodies belong to the \(L_{p}\)-Brunn-Minkowski theory, which is an extension of the classical Brunn-Minkowski theory. Apart from [13], \(L_{p}\)-projection bodies have been investigated intensively in recent years (see [6, 14–21]).
Through the characterization of so-called \(L_{p}\)-Minkowski valuations, Ludwig [15] discovered (see also [22–29] for related results) an asymmetric \(L_{p}\)-projection body \(\Pi^{+}_{p}K\) of \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\), whose support function is defined by
where \((u\cdot v)_{+}=\max\{u\cdot v, 0\}\). From (1.2) and (1.4) we see \(\Pi^{+}_{p}B=B\).
Moreover, Ludwig [15] introduced the function \(\varphi_{\tau}: \mathbb{R}\longrightarrow[0, +\infty)\) given by
for \(\tau\in[-1,1]\). For \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\), \(p\geq1\), let \(\Pi^{\tau}_{p}K\in{\mathcal {K}}^{n}_{\mathrm {o}}\) with support function
where
The normalization is chosen such that \(\Pi^{\tau}_{p}B=B\) for every \(\tau\in[-1,1]\). Here \(\Pi^{\tau}_{p}K\) is called the general \(L_{p}\)-projection body of K. Obviously, if \(\tau=0\), then \(\Pi^{\tau}_{p}K=\Pi^{0}_{p}K=\Pi_{p}K\).
Following the discovery of Ludiwg, Haberl and Schuster [30] defined
From (1.4), (1.5), and (1.6) they (see [30]) deduced that for \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\), \(p\geq1\), \(\tau\in[-1,1]\), and all \(u\in S^{n-1}\),
that is,
where ‘\(+_{p}\)’ denotes the \(L_{p}\)-Minkowski addition of convex bodies, and
If \(\tau=\pm1\), then \(\Pi^{\tau}_{p}K=\Pi^{\pm}_{p}K\).
For general \(L_{p}\)-projection bodies, Haberl and Schuster [30] not only established a general version of the \(L_{p}\)-Petty projection inequality but also determined the following extremum of volume for their polars.
Theorem 1.A
If \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\), \(p\geq 1\), and \(\tau\in[-1, 1]\), then
If K is not origin-symmetric and p is not an odd integer, then there is equality in the left inequality if and only if \(\tau=0\) and equality in the right inequality if and only if \(\tau=\pm1\). Here, \(\Pi^{\tau,\ast}_{p}K\) denotes the polar of the general \(L_{p}\)-projection body \(\Pi^{\tau}_{p}K\) of \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\).
Apart from [30], general \(L_{p}\)-projection bodies were studied by various authors; for example, Wang and Wan [31] investigated related Shephard-type problems, Wang and Feng [32] established Petty’s affine projection inequality for them. General \(L_{p}\)-projection bodies are a central notion in a new and rapidly evolving asymmetric \(L_{p}\)-Brunn-Minkowski theory (see [14, 15, 30–47]).
In this paper, we first extend inequality (1.10) to dual quermassintegrals forms, that is, the extremums of dual quermassintegrals for the polars of general \(L_{p}\)-projection bodies are obtained.
Theorem 1.1
If \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\), \(p\geq1\), \(\tau\in[-1, 1]\), and real \(i\neq n\), then, for \(i< n\) or \(i>n+p\),
and, for \(n< i< n+p\),
In each case, if K is not origin-symmetric and p is not an odd integer, then there is equality in the left inequality if and only if \(\tau=0\) and equality in the right inequality if and only if \(\tau=\pm1\). For \(i=n+p\), (1.11) and (1.12) become equalities. Here \(\widetilde{W}_{i}(Q)\) (i is any real) denote the dual quermassintegrals of the star body Q.
If \(i=0\), then since \(\widetilde{W}_{0}(Q)=V(Q)\), Theorem 1.1 reduces to Theorem 1.A.
Next, we obtain the extremums of quermassintegrals of general \(L_{p}\)-projection bodies.
Theorem 1.2
If \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\), \(p\geq1\), \(\tau\in[-1, 1]\), and \(i=0, 1, \ldots, n-1\), then
If K is not origin-symmetric and p is not an odd integer, then there is equality in the left inequality if and only if \(\tau=0\) and equality in the right inequality if and only if \(\tau=\pm1\). Here \(W_{i}(Q)\) (\(i=0, 1, \ldots, n-1\)) denote the quermassintegrals of \(Q\in{\mathcal {K}}^{n}_{\mathrm {o}}\).
Taking \(i=0\) in Theorem 1.2, we obtain the following:
Corollary 1.1
If \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\), \(p\geq 1\), and \(\tau\in[-1, 1]\), then
If K is not origin-symmetric and p is not an odd integer, then there is equality in the left inequality if and only if \(\tau=0\) and equality in the right inequality if and only if \(\tau=\pm1\).
Inequality (1.14) can be viewed as a dual version of inequality (1.10).
Finally, we determine the extremal values of the \(L_{q}\)-dual affine surface area (see Section 2) of the polars of general \(L_{p}\)-projection bodies.
Theorem 1.3
If \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\), \(p\geq 1\), \(0< q< n\), and \(\tau\in[-1, 1]\), then
If K is not origin-symmetric and p is not an odd integer, then there is equality in the left inequality if and only if \(\tau=0\) and equality in the right inequality if and only if \(\tau=\pm1\). Here \(\widetilde{\Omega}_{q}(Q)\) denotes the \(L_{q}\)-dual affine surface area of the star body Q.
This paper is organized as follows. In Section 2, we provide some preliminary results. Then, in Section 3, we recall some basic properties of general \(L_{p}\)-projection bodies. Section 4 contains the proofs of Theorems 1.1-1.3.
2 Basic notions
2.1 Radial functions and polar bodies
If K is a compact star-shaped (about the origin) set in \(\mathbb{R}^{n} \), then its radial function \(\rho_{K}=\rho(K,\cdot): \mathbb{R}^{n}\backslash\{0\}\longrightarrow[0,+\infty)\) is defined by (see [1])
If \(\rho_{K}\) is positive and continuous, then K is called a star body (about the origin). For the set of star bodies containing the origin in their interiors and the set of origin-symmetric star bodies in \(\mathbb{R}^{n}\), we write \({\mathcal {S}}^{n}_{\mathrm {o}}\) and \({\mathcal {S}}^{n}_{\mathrm{os}}\), respectively. Two star bodies K and L are said to be dilates (of one another) if \(\rho_{K}(u)/\rho_{L}(u)\) is independent of \(u\in S^{n-1}\).
If E is a nonempty subset of \(\mathbb{R}^{n}\), then the polar set \(E^{\ast}\) of E is defined by (see [1])
From (2.1) it follows that if \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\), then
2.2 \(L_{p}\)-Minkowski and \(L_{p}\)-harmonic radial combinations
For \(K, L\in{\mathcal {K}}^{n}_{\mathrm {o}}\), \(p\geq1\), and \(\lambda, \mu\geq0\) (not both zero), the \(L_{p}\)-Minkowski combination (also called the Firey \(L_{p}\)-combination) \(\lambda\cdot K+_{p}\mu\cdot L\in{\mathcal {K}}^{n}_{\mathrm {o}}\) of K and L is defined by (see [48, 49])
where ‘⋅’ in \(\lambda\cdot K\) denotes the \(L_{p}\)-Minkowski scalar multiplication.
For \(K, L\in\mathcal{S}^{n}_{\mathrm {o}}\), \(p\geq1\), and \(\lambda, \mu\geq0\) (not both zero), the \(L_{p}\)-harmonic radial combination \(\lambda\circ K\,\tilde{+}_{-p}\,\mu\circ L\in\mathcal{S}^{n}_{\mathrm {o}}\) of K and L is defined by (see [50])
From (2.2), (2.3), and (2.4) we easily see that if \(K, L\in{\mathcal {K}}^{n}_{\mathrm {o}}\), \(p\geq1\), and \(\lambda, \mu\geq0\) (not both zero), then
2.3 \(L_{p}\)-mixed and dual mixed volumes
Lutwak [51] gave the definition of \(L_{p}\)-mixed volume associated with \(L_{p}\)-Minkowski combinations of convex bodies: For \(K,L\in{\mathcal {K}}^{n}_{\mathrm {o}}\), \(\varepsilon>0\), and \(p\geq1\), the \(L_{p}\)-mixed volume \(V_{p}(K,L)\) of K and L is defined by
Corresponding to each \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\), Lutwak [51] proved that, for each \(L\in{\mathcal {K}}^{n}_{\mathrm {o}}\),
From (2.6) and (1.3) it follows immediately that, for each \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\),
The \(L_{p}\)-Minkowski inequality states the following (see [51]):
Theorem 2.A
If \(K, L\in\mathcal{K}^{n}_{\mathrm {o}}\), and \(p\geq1\), then
with equality for \(p>1\) if and only if K and L are dilates and for \(p=1\) if and only if K and L are homothetic.
Haberl [35] (also see [52]) introduced the notion of \(L_{p}\)-dual mixed volume as follows. For \({K, L}\in{\mathcal {S}}_{\mathrm {o}}^{n}\), \(p > 0\), and \(\varepsilon> 0\), the \(L_{p}\)-dual mixed volume \(\widetilde{V}_{p}(K, L)\) of K and L is defined by
where the integration is with respect to spherical Lebesgue measure on \(S^{n-1}\).
2.4 \(L_{p}\)-dual affine surface areas
Based on the \(L_{p}\)-dual mixed volume, Wang, Yuan, and He [53] defined the notion of \(L_{p}\)-dual affine surface area. For \(K\in {\mathcal {S}}_{\mathrm {o}}^{n}\) and \(0< p< n\), the \(L_{p}\)-dual affine surface area \(\widetilde{\Omega}_{p}(K)\) of K is defined by
where \({\mathcal {K}}_{c}^{n}\) denotes the set of convex bodies whose centroids lie at the origin in \(\mathbb{R}^{n}\).
We extend definition (2.10) from \(Q\in{\mathcal {K}}_{c}^{n}\) to \(Q\in {\mathcal {S}}_{\mathrm{os}}^{n}\) as follows: For \(K\in{\mathcal {S}}_{\mathrm {o}}^{n}\) and \(0< p< n\), the \(L_{p}\)-dual affine surface area \(\widetilde{\Omega }_{p}(K)\) of K is defined by
2.5 Quermassintegrals and dual quermassintegrals
For \(K\in\mathcal{K}^{n}\), \(i=0, 1, \ldots, n-1\), the quermassintegrals \(W_{i}(K)\) of K are given by (see [1, 49])
where \(S_{i}(K, \cdot)\) (\(i=0, 1, \ldots, n-1\)) denotes the ith surface area measure of K, and \(S_{0}(K, \cdot)=S(K, \cdot)\). From (2.12) and (2.7) we easily see that \(W_{0}(K) = V(K)\).
For the \(L_{p}\)-Minkowski combination, Lutwak [51] proved the following Brunn-Minkowski inequality for quermassintegrals.
Theorem 2.B
If \(K,L\in\mathcal{K}^{n}_{\mathrm {o}}\), \(p\geq1\), \(i=0, 1, \ldots, n-1\), and \(\lambda, \mu\geq0\) (not both zero), then
with equality for \(p=1\) if and only if K and L are homothetic and for \(p>1\) if and only if K and L are dilates.
For \(K\in\mathcal{S}^{n}_{\mathrm {o}}\) and any real i, the dual quermassintegrals \(\widetilde{W}_{i}(K)\) of K are defined by (see [54])
Obviously, (2.14) implies
Associated with the \(L_{p}\)-harmonic radial combinations of star bodies, Wang and Leng [55] established the following Brunn-Minkowski inequality for dual quermassintegrals.
Theorem 2.C
If \(K,L\in\mathcal{S}^{n}_{\mathrm {o}}\), \(p\geq1\), \(\lambda, \mu\geq0\) (not both zero), and real \(i\neq n\), then, for \(i< n\) or \(n< i< n+p\),
and, for \(i>n+p\),
In each inequality, equality holds if and only if K and L are dilates. For \(i=n+p\), (2.15) and (2.16) become equalities.
3 Some properties of general \(L_{p}\)-projection bodies
In this section, we recall some basic properties of general \(L_{p}\)-projection bodies.
Theorem 3.1
If \(K\in\mathcal{K}^{n}_{\mathrm {o}}\), \(p\geq1\), and \(\tau\in[-1, 1]\), then
Proof
From (1.5) it follows that, for all \(u\in S^{n-1}\),
This gives
In addition, by (1.9) we have that
From (3.3) and (3.4), together with (1.6) and (1.8), we obtain
Obviously, (3.2) and (3.5) yield (3.1). □
Theorem 3.2
If \(K\in\mathcal{K}^{n}_{\mathrm {o}}\), \(p\geq1\), \(\tau\in[-1, 1]\), and \(\tau\neq0\), then
Proof
From (1.8) and (3.4) it follows that, for \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\), \(p\geq1\), and \(\tau\in[-1,1]\),
that is,
for all \(u\in S^{n-1}\). Therefore, by (3.3), (1.7), and (3.6), if \(\Pi^{+}_{p}K=\Pi^{-}_{p}K\), then
for all \(u\in S^{n-1}\). This gives \(\Pi^{\tau}_{p}K=\Pi^{-\tau}_{p}K\).
Conversely, if \(\Pi^{\tau}_{p}K=\Pi^{-\tau}_{p}K\), then (1.7) and (3.6) yield
for all \(u\in S^{n-1}\). Since \(f_{1}(\tau)-f_{2}(\tau)\neq0\) when \(\tau\neq0\), we get \(\Pi^{+}_{p}K=\Pi^{-}_{p}K\). □
Haberl and Schuster [30] proved the following fact.
Theorem 3.A
If \(K\in\mathcal{K}^{n}_{\mathrm {o}}\), \(p\geq1\), and p is not odd integer, then \(\Pi^{+}_{p}K=\Pi^{-}_{p}K\) if and only if \(K\in\mathcal{K}^{n}_{\mathrm{os}}\).
According to Theorems 3.A and 3.2, we get the following:
Theorem 3.3
If \(K\in\mathcal{K}^{n}_{\mathrm {o}}\), \(p\geq1\), and p is not odd integer, then, for \(\tau\in[-1, 1]\) and \(\tau\neq0\), \(\Pi^{\tau}_{p}K=\Pi^{-\tau}_{p}K\) if and only if \(K\in\mathcal{K}^{n}_{\mathrm{os}}\).
Theorem 3.4
If \(K\in\mathcal{K}^{n}_{\mathrm {o}}\), \(p\geq1\), and \(\tau\in[-1, 1]\), then
Proof
From (1.7) and (3.6), using (3.3), we have that, for any \(u\in S^{n-1}\),
that is,
This is the desired relation. □
From Theorem 3.4 we deduce the following:
Corollary 3.1
If \(K\in\mathcal{K}^{n}_{\mathrm {o}}\), \(p\geq1\), and \(\tau\in[-1, 1]\), then
Proof
Taking \(\tau=0\) in (1.8) and combining with (1.9) yield
From (3.9) and (3.7) we immediately get (3.8). □
Theorem 3.5
If \(K, L\in\mathcal{K}^{n}_{\mathrm{os}}\), \(p\geq1\) is not an even integer, and \(\tau \in[-1, 1]\), then
The proof of Theorem 3.5 requires the following two lemmas.
Lemma 3.1
If \(K, L\in\mathcal{K}^{n}_{\mathrm {o}}\), and \(p\geq1\) is not an even integer, then \(\Pi _{p} K=\Pi_{p} L\) if and only if \(V_{p}(K, Q)=V_{p}(L, Q)\) for any \(Q\in\mathcal{K}^{n}_{\mathrm{os}}\).
Proof
From (1.1) we know that, for any \(u\in S^{n-1}\),
which implies \(\Pi_{p}(-K)=\Pi_{p}K\). Thus, for any \(u\in S^{n-1}\),
Thus, if \(\Pi_{p} K=\Pi_{p} L\), then, for any \(u\in S^{n-1}\),
Letting \(\mu(v)=S_{p}(K, v)+S_{p}(-K, v)-S_{p}(L, v)-S_{p}(-L, v)\), we have
Since \(\mu(v)\) is an even Borel measure on \(S^{n-1}\) and \(p\geq1\) is not an even integer, it follows from (3.10) that \(\mu(v)=0\) (see, e.g., [30]), that is,
Since \(Q\in\mathcal{K}^{n}_{\mathrm{os}}\), we have \(h_{Q}(-v)=h_{Q}(v)\) for all \(v\in S^{n-1}\). Therefore, by (2.6) we get
This and (2.6) yield
for any \(Q\in\mathcal{K}^{n}_{\mathrm{os}}\). By (3.11) we see that if \(\Pi_{p} K=\Pi_{p} L\), then \(V_{p}(K, Q)=V_{p}(L, Q)\) for any \(Q\in\mathcal{K}^{n}_{\mathrm{os}}\).
Conversely, if \(Q\in\mathcal{K}^{n}_{\mathrm{os}}\), let \(Q=[-u, u]\) (\(u\in S^{n-1}\)). Then \(h_{Q}(v)=| u\cdot v|\) for any \(v\in S^{n-1}\). This, together with (2.6), yields
Hence, if \(V_{p}(K, Q)=V_{p}(L, Q)\) for any \(Q\in\mathcal{K}^{n}_{\mathrm{os}}\), then \(\Pi_{p}K=\Pi_{p}L\). □
Lemma 3.2
If \(K, L\in\mathcal{K}^{n}_{\mathrm{os}}\) and \(p\geq1\) is not an even integer, then
Proof
By Lemma 3.1, if \(\Pi_{p} K=\Pi_{p} L\) and p is not an even integer, then, for any \(Q\in\mathcal{K}^{n}_{\mathrm{os}}\),
Taking K for Q in (3.12) and using (2.7) and (2.8), we obtain \(V(K)\geq V(L)\) 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). Similarly, taking L for Q in (3.12) yields \(V(K)\leq V(L)\), and equality holds for \(p>1\) if and only if K and L are dilates (for \(p=1\), if and only if K and L are homothetic). Therefore, \(V(K)=V(L)\), and K and L are dilates when \(p>1\) (K and L are homothetic when \(p=1\)). Since \(K, L\in\mathcal{K}^{n}_{\mathrm{os}}\), we have that, for \(p\geq1\), \(K=L\). □
Proof of Theorem 3.5
If \(K\in\mathcal{K}^{n}_{\mathrm{os}}\), then by (3.5) and Corollary 3.1 we have that
Therefore, if \(K, L\in\mathcal{K}^{n}_{\mathrm{os}}\), then, for \(\tau\in[-1, 1]\),
This, together with Lemma 3.2, completes the proof of Theorem 3.5. □
4 Proofs of the main results
In this section, we will complete the proofs of Theorems 1.1-1.3.
Proof of Theorem 1.1
Hence, for \(i< n\) or \(n< i< n+p\), using (4.1) and (2.15), we have that
But (3.1) yields \(\Pi^{-}_{p}K=\Pi^{+}_{p}(-K)=-\Pi^{+}_{p}K\), which implies \(\widetilde{W}_{i}(\Pi^{+,\ast}_{p}K)=\widetilde{W}_{i}(\Pi ^{-,\ast}_{p}K)\). Hence, by (4.2) and (3.3) we obtain
Now, if \(i< n\), then
Inequality (4.4) is just the right-hand side inequality of (1.11). If \(n< i< n+p\), then by (4.3) we get
which gives the right-hand side inequality of (1.12).
For \(i>n+p\), using (4.1) and (2.16), we arrive at
which yields (4.4).
According to the conditions of equality in (2.15) and (2.16), we have that equality holds in (4.4) and (4.5) if and only if \(\Pi^{+,\ast}_{p}K\) and \(\Pi^{-,\ast}_{p}K\) are dilates. From this, letting \(\Pi^{+,\ast}_{p}K=c\Pi^{-,\ast}_{p}K\) (\(c>0\)) and using that \(\widetilde{W}_{i}(\Pi^{+,\ast}_{p}K)=\widetilde{W}_{i}(\Pi^{-,\ast}_{p}K)\), it follows that \(c=1\), that is, \(\Pi^{+,\ast}_{p}K=\Pi^{-,\ast}_{p}K\). This means that \(\Pi^{+}_{p}K=\Pi^{-}_{p}K\). Hence, from Theorem 3.A we see that if K is not origin-symmetric and p is not an odd integer, then equality holds in the right-hand side inequalities of (1.11) and (1.12) if and only if \(\tau=\pm1\).
Now we prove the left-hand side inequalities of (1.11) and (1.12).
From (3.8) and (2.5) we have that
Using (3.1) and respectively combining with inequalities (2.15) and (2.16), we obtain the left-hand side inequalities of (1.11) and (1.12).
Moreover, by the conditions of equality in (2.15) and (2.16) we see that equality holds in the left-hand side inequalities of (1.11) and (1.12) if and only if \(\Pi^{\tau}_{p}K=\Pi^{-\tau}_{p}K\). This, together with Theorem 3.3, yields that if K is not origin-symmetric and p is not an odd integer, then equality holds in the left-hand side inequalities of (1.11) and (1.12) if and only if \(\tau=0\). □
Proof of Theorem 1.2
Using (1.8) and inequality (2.13), we have
which, combined with (3.3), yields
This gives the right-hand side inequality of (1.13).
According to the condition of equality in (2.13), we see that equality holds in the right-hand side inequality of (1.13) for \(p>1\) if and only if \(\Pi^{+}_{p}K\) and \(\Pi^{-}_{p}K\) are dilates (for \(p=1\), if and only if \(\Pi^{+}_{p}K\) and \(\Pi^{-}_{p}K\) are homothetic), which yields \(\Pi^{+}_{p}K=\Pi^{-}_{p}K\). Thus, from Theorem 3.A it follows that if K is not origin-symmetric and p is not an odd integer, then equality holds in the right-hand side inequality of (1.13) if and only if \(\tau=\pm1\).
Meanwhile, from (3.8) and inequality (2.13) we obtain
which, together with (3.1), yields
This is the left-hand side inequality of (1.13), where equality holds if and only if \(\Pi^{\tau}_{p}K=\Pi^{-\tau}_{p}K\). This, together with Theorem 3.3, shows that if K is not origin-symmetric and p is not an odd integer, then equality holds in the left-hand side inequality of (1.13) if and only if \(\tau=0\). □
The proof of Theorem 1.3 requires the following two lemmas.
Lemma 4.1
If \(K, L\in\mathcal{S}^{n}_{\mathrm {o}}\), \(0< q< n\), \(p\geq1\), and \(\lambda, \mu\geq 0\) (not both zero), then, for any \(Q\in\mathcal{S}^{n}_{\mathrm {o}}\),
with equality if and only if K and L are dilates.
Proof
Since \(0< q< n\) and \(p\geq1\), we have \(-p/(n-q)<0\). Hence, from (2.9), (2.4), and the Minkowski integral inequality (see [56]), we obtain that, for any \(Q\in\mathcal{S}^{n}_{\mathrm {o}}\),
Thus, inequality (4.7) is proven.
According to the equality condition of the Minkowski integral inequality, equality holds in (4.7) if and only if there exists a constant \(c>0\) such that
for any \(u\in S^{n-1}\), that is, K and L are dilates. □
Lemma 4.2
If \(K, L\in\mathcal{S}^{n}_{\mathrm {o}}\), \(0< q< n\), \(p\geq1\), and \(\lambda, \mu\geq 0\) (not both zero), then
with equality if and only if K and L are dilates.
Proof
For a bounded function \(\varphi>0\), we have
Thus, by (2.11), (4.7), and (4.9), noticing that \(-\frac{p}{n-q}<0\) when \(0< q< n\) and \(p\geq1\), we have that
This gives (4.8).
According to the equality condition of inequality (4.7), equality holds in inequality (4.8) if and only if K and L are dilates. □
Proof of Theorem 1.3
From (4.1) and (4.8) we have that, for \(0< q< n\) and \(p\geq1\),
But (2.9) shows that, for any \(Q\in{\mathcal {S}}_{\mathrm{os}}^{n}\), \(\widetilde {V}_{q}(-K, Q)=\widetilde{V}_{q}(K, Q)\). This and (2.11) give \(\widetilde {\Omega}_{q}(-K)=\widetilde{\Omega}_{q}(K)\). From this we see that
This, together with (4.10) and (3.3), yields
that is, for \(0< q< n\) and \(p\geq1\),
This is the right-hand side inequality (1.15).
According to the equality condition of inequality (4.8), equality holds in the right-hand side inequality of (1.15) if and only if \(\Pi ^{+,\ast}_{p}K\) and \(\Pi^{-,\ast}_{p}K\) are dilates. This and (4.11) give \(\Pi^{+,\ast}_{p}K=\Pi^{-,\ast}_{p}K\), that is, \(\Pi^{+}_{p}K=\Pi ^{-}_{p}K\). From this, by Theorem 3.A, it follows that if K is not origin-symmetric and p is not an odd integer, then equality holds in the right-hand side inequality of (1.15) if and only if \(\tau=\pm1\).
On the other hand, by (4.6) and inequality (4.8), noticing that
we obtain that, for \(0< q< n\), \(p\geq1\) and \(\tau\in[-1,1]\),
This yields the left-hand side inequality of (1.15).
According to the equality condition of (4.8) and using (4.12), we know that equality holds in the left-hand side inequality of (1.15) if and only if \(\Pi^{\tau}_{p}K=\Pi^{-\tau}_{p}K\). This, combined with Theorem 3.3, implies that if K is not origin-symmetric and p is not an odd integer, then equality holds in the left-hand side inequality of (1.15) if and only if \(\tau=0\). □
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The authors would like to sincerely thank the referees for very valuable and helpful comments and suggestions, which made the paper more accurate and readable. Research is supported in part by the Natural Science Foundation of China (Grant No. 11371224) and Foundation of Degree Dissertation of Master of China Three Gorges University (Grant No. 2014PY065).
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Wang, W., Wang, J. Extremum of geometric functionals involving general \(L_{p}\)-projection bodies. J Inequal Appl 2016, 135 (2016). https://doi.org/10.1186/s13660-016-1076-2
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DOI: https://doi.org/10.1186/s13660-016-1076-2