1 Introduction and main results
Let \(\mathcal{K}^{n}\) denote the set of convex bodies (compact, convex subsets with nonempty interiors) in 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 \(\mathcal{K}^{n}_{o}\) and \(\mathcal{K}^{n}_{c}\), respectively. \(S^{n}_{o}\) and \(S^{n}_{c}\), respectively, denote the set of star bodies (about the origin) and the set of origin-symmetric star bodies in \(\mathbb{R}^{n}\). Let \(S^{n-1}\) denote the unit sphere in \(\mathbb{R}^{n}\), and let \(V(K)\) denote the n-dimensional volume of a body K. For the standard unit ball B in \(\mathbb{R}^{n}\), we use \(\omega_{n} = V(B)\) to denote its volume.
The notion of geominimal surface area was discovered by Petty (see [
1]). For
\(K\in\mathcal{K}^{n}\), the geominimal surface area,
\(G(K)\), of
K is defined by
$$\omega_{n}^{\frac{1}{n}}G(K)= \inf\bigl\{ nV_{1}(K,Q)V \bigl(Q^{\ast}\bigr)^{\frac{1}{n}}: Q\in\mathcal{K}^{n}\bigr\} . $$
Here
\(Q^{\ast}\) denotes the polar of body
Q and
\(V_{1}(M, N)\) denotes the mixed volume of
\(M, N\in\mathcal{K}^{n}\) (see [
2]).
The geominimal surface area serves as a bridge connecting a number of areas of geometry: affine differential geometry, relative geometry, and Minkowskian geometry. Hence it receives a lot of attention (see,
e.g., [
3,
4]). Lutwak in [
5] showed that there were natural extensions of geominimal surface areas in the Brunn-Minkowski-Firey theory. It motivates extensions of some known inequalities for geominimal surface areas to
\(L_{p}\)-geominimal surface areas. The inequalities for
\(L_{p}\)-geominimal surface areas are stronger than their classical counterparts (see [
6‐
10]).
Based on
\(L_{p}\)-mixed volume, Lutwak [
5] introduced the notion of
\(L_{p}\)-geominimal surface area. For
\(K\in\mathcal{K}^{n}_{o}\),
\(p\geq1\), the
\(L_{p}\)-geominimal surface area,
\(G_{p}(K)\), of
K is defined by
$$\omega_{n}^{\frac{p}{n}}G_{p}(K)= \inf\bigl\{ nV_{p}(K,Q)V\bigl(Q^{\ast}\bigr)^{\frac{p}{n}}: Q\in \mathcal{K}^{n}_{o}\bigr\} . $$
Here
\(V_{p}(M, N)\) denotes the
\(L_{p}\)-mixed volume of
\(M, N\in\mathcal{K}^{n}_{o}\) (see [
5,
11]). Obviously, if
\(p=1\),
\(G_{p}(K)\) is just the geominimal surface area
\(G(K)\).
Recently, Wang and Qi [
12] introduced a concept of
\(L_{p}\)-dual geominimal surface area, which is a dual concept for
\(L_{p}\)-geominimal surface area and belongs to the dual
\(L_{p}\)-Brunn-Minkowski theory for star bodies also developed by Lutwak (see [
13,
14]). For
\(K\in S_{o}^{n}\), and
\(p\geq1\), the
\(L_{p}\)-dual geominimal surface area,
\(\widetilde{G}_{-p}(K)\), of
K is defined by
$$ \omega_{n}^{-\frac{p}{n}}\widetilde{G}_{-p}(K)= \inf\bigl\{ n \widetilde{V}_{-p}(K,Q)V\bigl(Q^{\ast}\bigr)^{-\frac{p}{n}}: Q\in \mathcal{K}_{c}^{n}\bigr\} . $$
(1.1)
Here,
\(\widetilde{V}_{-p}(M,N)\) denotes the
\(L_{p}\)-dual mixed volume of
\(M, N\in S_{o}^{n}\) (see [
5]).
Centroid bodies are a classical notion from geometry which have attracted increased attention in recent years (see [
13,
15‐
22]). In particular, Lutwak and Zhang [
18] introduced the notion of
\(L_{p}\)-centroid bodies. For each compact star-shaped (about the origin)
K in
\(\mathbb{R}^{n}\) and real number
\(p\geq1\), the
\(L_{p}\)-centroid body,
\(\Gamma_{p} K\), of
K is an origin-symmetric convex body whose support function is defined by
$$\begin{aligned} h^{p}_{\Gamma_{p} K}(u)&=\frac{1}{c_{n, p}V(K)}\int _{K}|u\cdot x|^{p} \,dx \\ &=\frac{1}{c_{n, p}(n+p)V(K)}\int_{S^{n-1}}|u\cdot v|^{p} \rho_{K}^{n+p}(v)\,dS(v) \end{aligned}$$
(1.2)
for all
\(u\in S^{n-1}\), where
$$ c_{n, p}=\omega_{n+p}/\omega_{2}\omega_{n} \omega_{p-1}, \quad\mbox{and}\quad \omega _{n}= \pi^{\frac{n}{2}}/\Gamma\biggl(1+\frac{n}{2}\biggr). $$
(1.3)
More recently, Feng
et al. [
23] defined a new notion of general
\(L_{p}\)-centriod bodies, which generalized the concept of
\(L_{p}\)-centroid bodies. For
\(K\in S_{o}^{n}\),
\(p\geq1\), and
\(\tau\in[-1, 1]\), the general
\(L_{p}\)-centroid body,
\(\Gamma_{p}^{\tau}K\), of
K is a convex body whose support function is defined by
$$\begin{aligned} h^{p}_{\Gamma_{p}^{\tau}K}(u)&=\frac{1}{c_{n, p}(\tau)V(K)}\int _{K}\varphi _{\tau}(u\cdot x)^{p} \,dx \\ & =\frac{1}{c_{n, p}(\tau)(n+p)V(K)}\int_{S^{n-1}}\varphi_{\tau}(u\cdot v)^{p}\rho_{K}^{n+p}(v)\,dv, \end{aligned}$$
(1.4)
where
$$c_{n, p}(\tau)=\frac{1}{2}c_{n, p}\bigl[(1+ \tau)^{p}+(1-\tau)^{p}\bigr], $$
and
\(\varphi_{\tau}: \mathbb{R}\rightarrow[0, \infty)\) is a function defined by
\(\varphi_{\tau}(t)=|t|+\tau t\). We note that general
\(L_{p}\)-centroid bodies are an essential part of the rapidly evolving asymmetric
\(L_{p}\)-Brunn-Minkowski theory (see [
20,
24‐
32]).
The normalization is chosen such that
\(\Gamma_{p}^{\tau}B=B\) for every
\(\tau\in[-1, 1]\), and
\(\Gamma_{p}^{0} K=\Gamma_{p} K\). Let
\(\varphi_{+}(u\cdot x)=\max\{u\cdot x, 0\}\) (
\(\tau=1\)) in (
1.4), then a special case of the definition of
\(\Gamma_{p}^{\tau}K\) is
\(\Gamma_{p}^{+} K\),
i.e.,
$$\begin{aligned} h^{p}_{\Gamma_{p}^{+} K}(u)&=\frac{1}{c_{n, p}V(K)}\int _{K}\varphi_{+}(u\cdot x)^{p} \,dx \\ &=\frac{1}{c_{n, p}(n+p)V(K)}\int_{S^{n-1}}\varphi_{+}(u\cdot v)^{p}\rho_{K}^{n+p}(v)\,dv. \end{aligned}$$
(1.5)
Besides, we also define
$$ \Gamma_{p}^{-}K=\Gamma_{p}^{+}(-K). $$
(1.6)
From the definition of
\(\Gamma^{\pm}_{p} K\) and (
1.4), we see that if
\(K\in S_{o}^{n}\),
\(p\geq1\), and
\(\tau\in[-1, 1]\), then
$$ \Gamma_{p}^{\tau}K=f_{1}(\tau)\cdot \Gamma_{p}^{+} K+_{p}f_{2}(\tau)\cdot \Gamma_{p}^{-} K, $$
(1.7)
where ‘
\(+_{p}\)’ denotes the Firey
\(L_{p}\)-combination of convex bodies, and
$$ f_{1}(\tau)=\frac{(1+\tau)^{p}}{(1+\tau)^{p}+(1-\tau)^{p}}, \qquad f_{2}(\tau )= \frac{(1-\tau)^{p}}{(1+\tau)^{p}+(1-\tau)^{p}}. $$
(1.8)
If
\(\tau=\pm1\) in (
1.7) and using (
1.8), then
$$\Gamma_{p}^{+1} K=\Gamma_{p}^{+} K, \qquad \Gamma_{p}^{-1} K=\Gamma_{p}^{-} K. $$
In [
16] Grinberg and Zhang discussed an investigation of Shephard type problems for
\(L_{p}\)-centriod bodies. Namely, let
K and
L be two origin-symmetric star bodies such that
$$\Gamma_{p} K\subset\Gamma_{p} L. $$
They proved that if the space
\((\mathbb{R}^{n}, \|\cdot\|_{L})\) embeds in
\(L_{p}\), then we necessarily have
On the other hand, if
\((\mathbb{R}^{n}, \|\cdot\|_{K})\) does not embed in
\(L_{p}\), then there is a body
L so that
\(\Gamma_{p} K\subset\Gamma_{p} L\), but
\(V(K)\leq V(L)\).
In this article, we first investigate the Shephard type problems for general \(L_{p}\)-centroid bodies and give the affirmative and negative parts of the version of \(L_{p}\)-dual geominimal surface area.
The proofs of Theorems
1.1-
1.3 will be given in Section
3.
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors completed the writing of this paper and read and approved the final manuscript.