On the sine polarity and the $L_{p}$-sine Blaschke-Santaló inequality ¹¹1Keywords: Blaschke-Santaló inequality, $L_{p}$-sine transform, $L_{p}$-sine centroid body, sine polar body.

Geometric inequalities, such as isoperimetric or affine isoperimetric inequalities, are the central objects of interest in convex geometry. These inequalities aim to estimate geometric (or affine) invariants from above and/or below in terms of the volume. One of the most important affine isoperimetric inequalities is the celebrated Blaschke-Santaló inequality. It asserts that, in the $n$-dimensional Euclidean space $\mathbb{R}^{n}$,

$$V(K)V(K^{\circ})\leq\omega_{n}^{2}$$

(1.1)

holds for any origin-symmetric convex body $K$ or more generally for any convex body $K$ with its centroid at the origin, with equality if and only if $K$ is an origin-symmetric ellipsoid (see, e.g., [4, 57, 56, 46, 51, 48, 47]). Here $V(\cdot)$ denotes the volume (Lebesgue measure) in $\mathbb{R}^{n}$ and the volume of the Euclidean unit ball $B^{n}$ in $\mathbb{R}^{n}$ is solely written by $\omega_{n}=\pi^{n/2}/\Gamma(1+n/2)$. A convex body $K$ in $\mathbb{R}^{n}$ is a convex compact subset of $\mathbb{R}^{n}$ with nonempty interior. If the convex body $K$ contains the origin in its interior, then the polar body $K^{\circ}$ of $K$ is defined by

$$K^{\circ}=\Big{\{}x\in\mathbb{R}^{n}:x\cdot y\leq 1\quad\textrm{for all}~{}y\in K\Big{\}},$$

where $x\cdot y$ is the inner product of $x,y\in\mathbb{R}^{n}$. In 1990’s, a more general version of the Blaschke-Santaló inequality was established by Lutwak and Zhang [43]. It states that, for $K\subset\mathbb{R}^{n}$ being a star body and $1\leq p\leq\infty$, one has

$$V(K)V(\Gamma_{p}^{\circ}K)\leq\omega_{n}^{2},$$

(1.2)

with equality if and only if $K$ is an origin-symmetric ellipsoid, where $\Gamma_{p}^{\circ}K$ denotes the polar body of the $L_{p}$ centroid body $\Gamma_{p}K$ whose support function at $x\in\mathbb{R}^{n}$ is given by

$$h_{\Gamma_{p}K}(x)^{p}=\frac{1}{c_{n,p}V(K)}\int_{K}|x\cdot y|^{p}\,dy,$$

(1.3)

where $\,dy$ is the Lebesgue measure and

$$c_{n,p}=\frac{\omega_{n+p}}{\omega_{2}\omega_{n}\omega_{p-1}},$$

(1.4)

with $\omega_{p}=\pi^{n/2}/\Gamma(1+p/2)$ for $p>0$. The normalization above is chosen so that $\Gamma_{p}B^{n}=B^{n}$. The body $\Gamma_{\infty}K$ is to be interpreted as the limit of $\Gamma_{p}K$ for $p\rightarrow\infty$. In the case that $K$ is an origin-symmetric convex body and $p=\infty$, $\Gamma_{\infty}^{\circ}K$ coincides with $K^{\circ}$, and hence inequality (1.2) becomes the Blaschke-Santaló inequality (1.1). So inequality (1.2) is usually called the $L_{p}$ Blaschke-Santaló inequality.

It can be observed that the definition of the $L_{p}$ centroid body is based on the cosine function (i.e., the inner product). More specifically, integrating in polar coordinates, for $x\in\mathbb{R}^{n}$,

$$h_{\Gamma_{p}K}(x)^{p}=\!\frac{n\omega_{n}}{(n+p)c_{n,p}V(K)}\!\int_{S^{n-1}}|x\cdot u|^{p}\rho_{K}(u)^{n+p}\,du=\frac{n\omega_{n}}{(n+p)c_{n,p}V(K)}\big{(}\mathcal{C}_{p}\,\rho_{K}(\cdot)^{n+p}\big{)}(x),$$

where $\,du$ is the rotation invariant probability measure on the unit sphere $S^{n-1}$ in $\mathbb{R}^{n}$ and $\mathcal{C}_{p}\,\mu$ denotes the $L_{p}$-cosine transform of a Borel measure $\mu$ defined on $S^{n-1}$. That is, for $p>0$,

$$(\mathcal{C}_{p}\,\mu)(x)=\int_{S^{n-1}}|x\cdot u|^{p}\,d\mu(u),\ \ \ x\in\mathbb{R}^{n}.$$

(1.5)

The $L_{p}$-cosine transform provides a very useful analytical operator for convex geometry and plays a dominating role in applications, see, e.g., the books [13, 26, 58, 24] and references [6, 18, 19, 29, 31, 35, 39, 40, 43, 55, 30, 59], among others.

Similar to (1.5), $\mathcal{S}_{p}\,\mu$, the $L_{p}$-sine transform of a Borel measure $\mu$ defined on $S^{n-1}$, can be defined for $p>0$ as follows:

$$(\mathcal{S}_{p}\,\mu)(x)=\int_{S^{n-1}}|\textrm{P}_{u^{\perp}}x|^{p}\,d\mu(u)=\int_{S^{n-1}}[x,u]^{p}\,d\mu(u),\ \ \ x\in\mathbb{R}^{n},$$

(1.6)

where $\textrm{P}_{u^{\perp}}x$ is the orthogonal projection of $x$ onto the $(n-1)$-dimensional subspace $u^{\perp}$ perpendicular to $u$, and $[x,u]=\sqrt{|x|^{2}|u|^{2}-|x\cdot u|^{2}}$ with $|x|$ being the Euclidean norm of $x\in\mathbb{R}^{n}$. Note that $[x,u]$ represents the sine function due to the Pythagoras theorem, and geometrically $[x,u]$ is the $2$-dimensional volume of the parallelepiped spanned by $x$ and $u$. The sine, cosine, and Radon transforms are closely related and have been extensively studied, see, e.g., [16, 52, 54, 53, 45, 60, 14, 15, 20]. The $L_{p}$-sine transform (i.e., $m=n-1$) and the $L_{p}$-cosine transform (i.e., $m=1$) on $S^{n-1}$ are the special cases of the transform $\mathcal{C}_{m,p}\,\mu$ on Grassmann manifolds $\mathbf{G}_{n,m}$ (the set of $m$-dimensional linear subspaces in $\mathbb{R}^{n}$) [28]. Here, for a Borel measure $\mu$ on $\mathbf{G}_{n,m}$ and $p>0$,

$$(\mathcal{C}_{m,p}\,\mu)(x)=\int_{\mathbf{G}_{n,m}}|\textrm{P}_{\xi}x|^{p}d\mu(\xi),\ \ \ x\in\mathbb{R}^{n}.$$

It can be checked that, when $p\geq 1$, $(\mathcal{C}_{m,p}\,\mu)^{1/p}$ defines a norm on $\mathbb{R}^{n}$ and hence its unit ball is an origin-symmetric convex body in $\mathbb{R}^{n}$. Some isoperimetric and reverse isoperimetric inequalities for the unit balls induced by the norm $(\mathcal{C}_{m,p}\,\mu)^{1/p}$ were established in [28]. Based on the $L_{2}$-sine transform, the first three authors of the present paper proposed two new sine ellipsoids, and obtained sharp volume inequalities and valuation properties for these sine ellipsoids in [27]. Note that these sine ellipsoids are closely related in the Pythagorean relation and duality to the Legendre ellipsoid and its dual ellipsoid (known as the LYZ ellipsoid) [36]. It is worth mentioning that, when $n=2$, the $L_{p}$-sine transform coincides with the $L_{p}$-cosine transform up to a rotation of $\pi/2$. Thus, properties for the $L_{p}$-cosine transform shall hold for the $L_{p}$-sine transform on $\mathbb{R}^{2}$. Unfortunately, this is no long true when $n\geq 3$, and the situation is totally different. The main difficulty for the $L_{p}$-sine transform is the lack of affine nature, which makes the analysis on the $L_{p}$-sine transform often more challenging.

The main goal of the present paper is to establish the $L_{p}$-sine Blaschke-Santaló inequality (1.9), which can be viewed as the “sine cousin” of the $L_{p}$ Blaschke-Santaló inequality (1.2). Furthermore, a sine version of the Blaschke-Santaló inequality (1.10) with characterization of equality is also obtained. Thus, our results are complementary to the classical studies for the cosine transform.

In Section 3, we introduce a new convex body in $\mathbb{R}^{n}$ (i.e., the $L_{p}$-sine centroid body $\Lambda_{p}K$ for $K$ being an $L_{n+p}$-star) in terms of the $L_{p}$-sine transform (1.6), whose support function at $x\in\mathbb{R}^{n}$ is defined by, for $p\geq 1$,

$$h_{\Lambda_{p}K}(x)^{p}=\frac{n\omega_{n}}{(n+p)\widetilde{c}_{n,p}V(K)}\big{(}\mathcal{S}_{p}\,\rho_{K}(\cdot)^{n+p}\big{)}(x)=\frac{1}{\widetilde{c}_{n,p}V(K)}\int_{K}[x,y]^{p}\,dy,$$

(1.7)

where

$$\widetilde{c}_{n,p}=\frac{(n-1)\omega_{n-1}\omega_{n+p-2}}{(n+p)\omega_{n}\omega_{n+p-3}}.$$

(1.8)

In particular, $\Lambda_{p}B^{n}=B^{n}$ (see (4.1)). Note that $\Lambda_{p}K$ is well-defined since $h_{\Lambda_{p}K}$ is a convex function (see Section 3 for details). In the special case when $K$ is a star body in $\mathbb{R}^{n}$, $\Lambda_{p}K$ was first proposed in [27]. Motivated by the limit of $\Lambda_{p}^{\circ}K$ as $p\rightarrow\infty$, the sine polar body of $K$, denoted by $K^{\diamond}$, is introduced. Namely, for a subset $K\subset\mathbb{R}^{n}$, we let

$$K^{\diamond}=\Big{\{}x\in\mathbb{R}^{n}:[x,y]\leq 1\ \ \mathrm{for\ all}\ y\in K\Big{\}}.$$

The sine polar body $K^{\diamond}$ differs from $K^{\circ}$ mainly with $[x,y]$ replacing $x\cdot y$. Hence many properties for $K^{\diamond}$ are similar to those for $K^{\circ}$, for instance, $L^{\diamond}\subseteq K^{\diamond}$ if $K\subseteq L$, $(B^{n})^{\diamond}=B^{n}$, and $(cK)^{\diamond}=c^{-1}\cdot K^{\diamond}$ for any $c>0$. However, some properties for $K^{\diamond}$ are completely different from those for $K^{\circ}$. For instance, the volume product $V(K)V(K^{\circ})$ is $\textrm{GL}(n)$-invariant (i.e., $V(\phi K)V((\phi K)^{\circ})=V(K)V(K^{\circ})$ for any invertible linear map $\phi$ on $\mathbb{R}^{n}$), but the volume product $V(K)V(K^{\diamond})$ is in general not $\textrm{GL}(n)$-invariant (except for $n=2$). The non-$\textrm{GL}(n)$-invariance for $V(K)V(K^{\diamond})$ does bring extra difficulty in characterizing the equality for the sine Blaschke-Santaló inequality (1.10) in Section 5. On the other hand, the sine bipolar property $K=K^{\diamond\diamond}$, where $K^{\diamond\diamond}=(K^{\diamond})^{\diamond}$, does not hold in general for all convex bodies. Indeed, the definition of $K^{\diamond}$ indicates that $K^{\diamond}$ is formed by the intersection of closed solid cylinders (see (3.7)). This brings our attention to a special class of convex bodies $\mathcal{C}_{e}^{n}$ consisting of all origin-symmetric convex bodies generated by the intersection of origin-symmetric closed solid cylinders. The sine bipolar property $K=K^{\diamond\diamond}$ holds for all $K\in\mathcal{C}_{e}^{n}$, see Proposition 3.3 (v). Proposition 3.5 (ii) also shows that $K^{\diamond\diamond}$ is just the cylindrical hull of $K$ – the intersection of all origin-symmetric closed solid cylinders containing $K$. The cylindrical hull of $K$ is a notion analogous to the convex hull, and any convex body $K\in\mathcal{C}_{e}^{n}$ must be equal to the cylindrical hull of itself (see Proposition 3.5 (i)). Moreover, in Section 3, we develop many notions in $\mathcal{C}_{e}^{n}$, such as the cylindrical support function, the supporting cylinder, and the cylindrical Gauss image. These notions are completely parallel to their classical counterparts, e.g., the support function, the supporting hyperplane, and the Gauss image.

Last but not the least, we would like to mention that the notion $\mathcal{C}_{e}^{n}$ is quite important. A basic example of $\mathcal{C}_{e}^{n}$ is the intersection of two closed solid cylinders with equal radius at right angles. The volume of this bicylinder (also known as “Mou He Fang Gai” in China) in $\mathbb{R}^{3}$ was first studied by Archimedes more than 2200 years ago and independently by the old Chinese mathematicians Hui Liu, Chongzhi Zu and Geng Zu more than 1500 years ago (see, e.g., [23, 25]). An amazing application of the bicylinder is the calculation of the volume of $3$-dimensional balls by these old Chinese mathematicians [25]. The solids obtained as the intersection of two or three cylinders of equal radius at right angles are better known as the Steinmetz solids in Europe, see Figure 1. Applications of the Steinmetz solid are common in real life, such as the T-adapter and the circular pipes joining. The intersection of cylinders also naturally appears in the study of crystal (see, e.g., [1, 49]).

In Section 4, we will prove the following $L_{p}$-sine Blaschke-Santaló inequality: let $p\geq 1$ and $K\subset\mathbb{R}^{n}$ be a star body. Then

$$V(K)V(\Lambda_{p}^{\circ}K)\leq\omega_{n}^{2},$$

(1.9)

with equality if and only if $K$ is an origin-symmetric ellipsoid when $n=2$, and is an origin-symmetric ball when $n\geq 3$. For $p=2$, this inequality was also established by the first three authors [27, Theorem 4.1]. By letting $p\rightarrow\infty$, the $L_{p}$-sine Blaschke-Santaló inequality will lead to the following sine Blaschke-Santaló inequality: let $K\subset\mathbb{R}^{n}$ be a star body. Then

$$V(K)V(K^{\diamond})\leq\omega_{n}^{2},$$

(1.10)

with equality if and only if $K$ is an origin-symmetric ellipsoid when $n=2$, and is an origin-symmetric ball when $n\geq 3$.

Therefore, inequalities (1.9) and (1.10) can be viewed as the “sine cousin” of inequalities (1.2) and (1.1). However, the approximation as $p\rightarrow\infty$ does not yield the equality conditions of inequality (1.10). To characterize these equality conditions requires a lot of work mainly because of the extra difficulty arising from the non-$\textrm{GL}(n)$-invariance of $V(K)V(K^{\diamond})$ for $n\geq 3$. In this sense, equality conditions in inequality (1.1) and its sine counterpart (1.10) are different for $n\geq 3$. However, inequalities (1.1) and (1.10) are in fact equivalent if $K$ is an origin-symmetric convex body in $\mathbb{R}^{2}$. Indeed, it will be shown in Section 4 that under this circumstance $K^{\diamond}$ is just a rotation of $K^{\circ}$ by an angle of $\pi/2$. A similar situation also happens to (1.2) and (1.9).

We also mention that information theory (or probability theory) and convex geometry are closely related (see, e.g., [12, Section 14]). In recent years there was a growing body of works in this direction, see, e.g., [5, 9, 11, 17, 37, 38, 41, 42, 33, 34, 50, 44]. For example, an important application of the $L_{p}$ Blaschke-Santaló inequality (1.2) to information theory leads to the celebrated $L_{p}$ moment-entropy inequality (see (6.7)) due to Lutwak, Yang, and Zhang [38]. Similarly, the “sine cousin” of $L_{p}$ moment-entropy inequality (see (6.16)) can be deduced by using the $L_{p}$-sine Blaschke-Santaló inequality. We believe that the applications of the $L_{p}$-sine transform to information theory (or probability theory) will likely produce more interesting and useful results for applications in many areas.

Throughout the paper, $\mathbb{R}^{n}$ denotes $n$-dimensional Euclidean space ($n\geq 2$) and the term subspace means a linear subspace. As usual, $x\cdot y$ denotes the inner product of $x,y\in\mathbb{R}^{n}$ and $|x|$ the Euclidean norm of $x$. The unit sphere $\{x\in\mathbb{R}^{n}:|x|=1\}$ is denoted by $S^{n-1}$ and the unit ball $\{x\in\mathbb{R}^{n}:|x|\leq 1\}$ by $B^{n}$. Furthermore, we denote by $o$ the origin in $\mathbb{R}^{n}$ and $\bar{x}=x/|x|\in S^{n-1}$ for all $x\in\mathbb{R}^{n}\setminus\{o\}$. Note that the cosine of the angle between nonzero vectors $x$ and $y$ is indeed given by $\bar{x}\cdot\bar{y}$. Thus, for $x,y\in\mathbb{R}^{n}\setminus\{o\}$,

$$|x\cdot y|=|x|\cdot|y|\cdot|\bar{x}\cdot\bar{y}|=|x|\cdot|\textrm{P}_{x}y|,$$

(2.1)

where $\textrm{P}_{x}y$ is the orthogonal projection of $y$ onto the $1$-dimensional subspace containing $x$. On the other hand, the sine of the angle between nonzero vectors $x$ and $y$ can be easily calculated through the formula $\sqrt{1-|\bar{x}\cdot\bar{y}|^{2}}$. Corresponding to (2.1), for $x,y\in\mathbb{R}^{n}\setminus\{o\}$, let

$$[x,y]:=|x|\cdot|y|\cdot\sqrt{1-|\bar{x}\cdot\bar{y}|^{2}}=\sqrt{|x|^{2}|y|^{2}-|x\cdot y|^{2}}=|x|\cdot\sqrt{|y|^{2}-|\textrm{P}_{x}y|^{2}}=|x|\cdot|\textrm{P}_{x^{\perp}}y|,$$

(2.2)

where $\textrm{P}_{x^{\perp}}y$ is the orthogonal projection of $y$ onto the $(n-1)$-dimensional subspace $x^{\perp}$ perpendicular to $x$. In other words, $[x,y]$ represents the $2$-dimensional volume of the parallelepiped spanned by $x$ and $y$. Of course, one can switch the roles of $x,y$ to get $[x,y]=[y,x]=|y|\cdot|\textrm{P}_{y^{\perp}}x|$. Moreover, if one or both of $x$ and $y$ are zero vectors, one can simply let $[x,y]=0$.

A convex body in $\mathbb{R}^{n}$ is a convex compact subset of $\mathbb{R}^{n}$ with nonempty interior. By $\mathscr{K}_{o}^{n}$ we mean the class of all convex bodies with the origin $o$ in their interiors. For each $K\in\mathscr{K}_{o}^{n}$, one can define its support function $h_{K}:\mathbb{R}^{n}\rightarrow(0,\infty)$ by

$$h_{K}(x)=\max\{x\cdot y:y\in K\},\ \ \ x\in\mathbb{R}^{n}.$$

(2.3)

A convex body $K\in\mathscr{K}_{o}^{n}$ is said to be origin-symmetric if $-x\in K$ for all $x\in K$. Denote by $\mathscr{K}_{e}^{n}$ the subclass of all origin-symmetric elements of $\mathscr{K}_{o}^{n}$.

Denote by $\textrm{GL}(n)$ the group of all invertible linear transforms on $\mathbb{R}^{n}$ and $\textrm{SL}(n)$ the subgroup of $\textrm{GL}(n)$ whose determinant is $1$. By $\textrm{O}(n)$ we mean the orthogonal group of $\mathbb{R}^{n}$, that is, the group of linear transformations preserving the inner product. For each $\phi\in\textrm{GL}(n)$, let $\phi^{t}$ and $\phi^{-1}$ be the transpose and inverse of $\phi$, respectively. For $\phi\in\textrm{GL}(n)$ and $x\in\mathbb{R}^{n}$, let $\phi K=\{\phi y:y\in K\}$ and then

$$h_{\phi K}(x)=h_{K}(\phi^{t}x).$$

(2.4)

Throughout the paper, $\,du$ denotes the rotation invariant probability measure on $S^{n-1}$. Let $q>0$. We say that a function $\rho:S^{n-1}\rightarrow[0,\infty)$ is $q$-integrable on $S^{n-1}$ if $\rho$ is measurable and

$$\int_{S^{n-1}}\rho(u)^{q}\,du<\infty.$$

Denote by $L^{q}(S^{n-1})$ the set of all $q$-integrable functions on $S^{n-1}$. As $\int_{S^{n-1}}\,du=1$, one can easily obtain, by using the Hölder inequality, that if $0<q_{1}<q$, then

$$L^{q}(S^{n-1})\subseteq L^{q_{1}}(S^{n-1}).$$

(2.5)

A set $L\subset\mathbb{R}^{n}$ is said to be star-shaped about the origin if $o\in L$ and

$$L=\big{\{}ru:0\leq r\leq\rho(u)\ \ \textrm{for}\ \ u\in S^{n-1}\big{\}},$$

where $\rho:S^{n-1}\rightarrow[0,\infty)$ is a nonnegative function. Such a function $\rho$ is called the radial function of $L$, which will be more often written as $\rho_{L}.$ The radial function can be extended to $\rho_{L}:\mathbb{R}^{n}\setminus\{o\}\rightarrow[0,\infty)$ by $\rho_{L}(ru)=r^{-1}\rho_{L}(u)$ for $r>0$ and $u\in S^{n-1}$, which gives

$$\rho_{L}(x)=\max\big{\{}\lambda\geq 0:\lambda x\in L\big{\}},\ \ \ x\in\mathbb{R}^{n}\setminus\{o\}.$$

(2.6)

Let $p>0$. A star-shaped set $L$ in $\mathbb{R}^{n}$ is said to be an $L_{n+p}$-star generated by $\rho\in L^{n+p}(S^{n-1})$ if $\rho_{L}=\rho$. Clearly, it follows from (2.6) that for $c>0$ and $x\in\mathbb{R}^{n}\backslash\{o\}$,

$$\rho_{cL}(x)=c\rho_{L}(x).$$

(2.7)

For $\phi\in\textrm{GL}(n)$ and $x\in\mathbb{R}^{n}\backslash\{o\}$, one has $\rho_{\phi L}(x)=\rho_{L}(\phi^{-1}x)$.

A set $L\subset\mathbb{R}^{n}$ is called a star body if $L$ is star-shaped and $\rho_{L}$ restricted on $S^{n-1}$ is positive and continuous. Denote by $\mathcal{S}^{n}$ the class of all star bodies. Clearly, $\mathscr{K}_{o}^{n}\subset\mathcal{S}^{n}$, and $L$ is an $L_{n+p}$-star for all $p>0$ if $L\in\mathcal{S}^{n}$. Moreover, the Minkowski functional $\|\cdot\|_{L}:\mathbb{R}^{n}\rightarrow[0,\infty)$ of $L\in\mathcal{S}^{n}$ is defined by

$$\|x\|_{L}=\min\{\lambda\geq 0:x\in\lambda L\},\ \ \ x\in\mathbb{R}^{n}.$$

(2.8)

Recall that the polar body $K^{\circ}$ of $K\in\mathscr{K}_{o}^{n}$ is defined by

$$K^{\circ}=\Big{\{}x\in\mathbb{R}^{n}:x\cdot y\leq 1\quad\textrm{for all}~{}y\in K\Big{\}}.$$

(2.9)

It can be easily verified that for $K\in\mathscr{K}_{o}^{n}$,

	$\displaystyle(cK)^{\circ}\!\!\!$	$\displaystyle=$	$\displaystyle\!\!\!c^{-1}K^{\circ},\ \ \ \mathrm{for\ any}\ c>0,$		(2.10)
	$\displaystyle(\phi K)^{\circ}\!\!\!$	$\displaystyle=$	$\displaystyle\!\!\!\phi^{-t}K^{\circ},\ \ \ \mathrm{for\ any}\ \phi\in\textrm{GL}(n).$		(2.11)

Moreover, it follows from (2.3), (2.6), (2.8), and (2.9) that if $K\in\mathscr{K}_{o}^{n}$,

$$\|x\|_{K^{\circ}}=1/\rho_{K^{\circ}}(x)=h_{K}(x),\ \ \ \mathrm{for\ any}\ x\in\mathbb{R}^{n}\setminus\{o\}.$$

(2.12)

In particular, $(K^{\circ})^{\circ}=K$ for any $K\in\mathscr{K}_{o}^{n}$. Note that if $K\in\mathscr{K}_{e}^{n}$ is an origin-symmetric convex body in $\mathbb{R}^{n}$, (2.9) can also be rewritten as

$$K^{\circ}=\{x\in\mathbb{R}^{n}:|x\cdot y|\leq 1\quad\textrm{for all}~{}y\in K\}.$$

(2.13)

Let $p>0$. For $L$ being an $L_{n+p}$-star, its volume is given by

$$V(L)=\omega_{n}\int_{S^{n-1}}\rho_{L}(u)^{n}\,du.$$

(2.14)

By (2.5), for an $L_{n+p}$-star $L$, it has a finite volume as $\rho_{L}\in L^{n+p}(S^{n-1}).$ Moreover, the $L_{-p}$ dual mixed volume [38] of an $L_{n+p}$-star $L$ and a star body $L^{\prime}\in\mathcal{S}^{n}$ can be formulated by

$$\widetilde{V}_{-p}(L,L^{\prime})=\omega_{n}\int_{S^{n-1}}\rho_{L}(u)^{n+p}\rho_{L^{\prime}}(u)^{-p}\,du.$$

(2.15)

In particular, for $L\in\mathcal{S}^{n}$ and $p>0$, one has

$$\widetilde{V}_{-p}(L,L)=V(L).$$

(2.16)

Following from the Hölder inequality, one can easily obtain the following dual $L_{-p}$ Minkowski inequality [38]: for $p>0$,

$$\widetilde{V}_{-p}(L,L^{\prime})^{n}\geq V(L)^{n+p}V(L^{\prime})^{-p}$$

(2.17)

holds for any $L_{n+p}$-star $L$ and any $L^{\prime}\in\mathcal{S}^{n}$, with equality if and only if there exists a constant $c>0$ such that $\rho_{L^{\prime}}(u)=c\cdot\rho_{L}(u)$ for almost all $u\in S^{n-1}$ with respect to the spherical measure of $S^{n-1}$.

Let $p\geq 1$ and $n\geq 2$. We first show that, under the assumption that $K\subset\mathbb{R}^{n}$ is an $L_{n+p}$-star with $V(K)>0$, the set $\Lambda_{p}K$ defined in (1.7) is well-defined. Indeed, we can get $h_{\Lambda_{p}K}\in(0,\infty)$ on $\mathbb{R}^{n}\setminus\{o\}$. To see this, for any $x\in\mathbb{R}^{n}\setminus\{o\}$ and any $y\in K$, (2.2) implies $[x,y]\leq|x|\cdot|y|$ and hence

	$\displaystyle h_{\Lambda_{p}K}(x)^{p}$	$\displaystyle=\frac{1}{\widetilde{c}_{n,p}V(K)}\int_{K}[x,y]^{p}\,dy$
		$\displaystyle\leq\frac{1}{\widetilde{c}_{n,p}V(K)}\int_{K}|x|^{p}|y|^{p}\,dy$
		$\displaystyle=\frac{n\omega_{n}|x|^{p}}{(n+p)\widetilde{c}_{n,p}V(K)}\int_{S^{n-1}}\rho_{K}(u)^{n+p}\,du<\infty.$

On the other hand, as $V(K)>0$, then $K$ cannot be concentrated on a $1$-dimensional space, and $[x,y]>0$ for almost all $y\in K$ (with respect the Lebesgue measure). Hence, $h_{\Lambda_{p}K}(x)>0$ for all $x\in\mathbb{R}^{n}\setminus\{o\}$.

It can be easily verified that $h_{\Lambda_{p}K}(tx)=t\cdot h_{\Lambda_{p}K}(x)$ for all $t\geq 0$ and $x\in\mathbb{R}^{n}$. Moreover, $h_{\Lambda_{p}K}(x)$ is a convex function, namely for all $\tau\in[0,1]$ and $x,x^{\prime}\in\mathbb{R}^{n}$, one has

$$h_{\Lambda_{p}K}(\tau x+(1-\tau)x^{\prime})\leq\tau\cdot h_{\Lambda_{p}K}(x)+(1-\tau)\cdot h_{\Lambda_{p}K}(x^{\prime}).$$

To see this, for $y\in\mathbb{R}^{n}\setminus\{o\}$, one has $[x,y]=|y|\cdot|\textrm{P}_{y^{\perp}}x|$ and hence

	$\displaystyle[\tau x+(1-\tau)x^{\prime},y]\!\!$	$\displaystyle=$	$\displaystyle\!\!|y|\cdot\big{|}\textrm{P}_{y^{\perp}}\big{(}\tau x+(1-\tau)x^{\prime}\big{)}\big{|}$		(3.1)
		$\displaystyle=$	$\displaystyle\!\!|y|\cdot\big{|}\tau\textrm{P}_{y^{\perp}}x+(1-\tau)\textrm{P}_{y^{\perp}}x^{\prime}\big{|}$
		$\displaystyle\leq$	$\displaystyle\!\!\tau|y|\cdot\big{|}\textrm{P}_{y^{\perp}}x\big{|}+(1-\tau)|y|\cdot\big{|}\textrm{P}_{y^{\perp}}x^{\prime}\big{|}$
		$\displaystyle=$	$\displaystyle\!\!\tau[x,y]+(1-\tau)[x^{\prime},y].$

Clearly, (3.1) is also valid for $y=o$. It then follows from the Minkowski inequality that

	$\displaystyle h_{\Lambda_{p}K}(\tau x+(1-\tau)x^{\prime})\!\!$	$\displaystyle=$	$\displaystyle\!\!\bigg{(}\frac{1}{\widetilde{c}_{n,p}V(K)}\int_{K}[\tau x+(1-\tau)x^{\prime},y]^{p}\,dy\bigg{)}^{1/p}$
		$\displaystyle\leq$	$\displaystyle\!\!\bigg{(}\frac{1}{\widetilde{c}_{n,p}V(K)}\int_{K}\Big{(}\tau[x,y]+(1-\tau)[x^{\prime},y]\Big{)}^{p}\,dy\bigg{)}^{1/p}$
		$\displaystyle\leq$	$\displaystyle\!\!\tau\cdot\bigg{(}\frac{1}{\widetilde{c}_{n,p}V(K)}\int_{K}[x,y]^{p}\,dy\bigg{)}^{1/p}+(1-\tau)\cdot\bigg{(}\frac{1}{\widetilde{c}_{n,p}V(K)}\int_{K}[x^{\prime},y]^{p}\,dy\bigg{)}^{1/p}$
		$\displaystyle=$	$\displaystyle\!\!\tau\cdot h_{\Lambda_{p}K}(x)+(1-\tau)\cdot h_{\Lambda_{p}K}(x^{\prime}).$

Furthermore, $\Lambda_{p}K$ is clearly origin-symmetric. We conclude that $\Lambda_{p}K\in\mathscr{K}_{e}^{n}$ for $p\geq 1$.

Denote by $\Lambda_{p}^{\circ}K$ the polar body of $\Lambda_{p}K$. It follows from (2.12), (1.7), and the polar coordinate that

$\displaystyle\rho_{\Lambda_{p}^{\circ}K}(x)^{-p}=\frac{n\omega_{n}}{(n+p)\widetilde{c}_{n,p}V(K)}\int_{S^{n-1}}[x,u]^{p}\rho_{K}(u)^{n+p}\,du,\ \ \ x\in\mathbb{R}^{n}\setminus\{o\}.$

(3.2)

By (2.7) and (3.2), one has

$$\Lambda_{p}^{\circ}(cK)=c^{-1}\Lambda_{p}^{\circ}K,\ \ \ \mathrm{for\ any}\ c>0.$$

(3.3)

If $K\in\mathcal{S}^{n}$ is a star body in $\mathbb{R}^{n}$, then $K$ is an $L_{n+p}$-star for all $p\geq 1$ with $V(K)>0$. Thus, by (1.7) and Stirling’s formula,

$$h_{\Lambda_{\infty}K}(x)=\lim_{p\rightarrow\infty}h_{\Lambda_{p}K}(x)=\max_{y\in K}\,[x,y],\ \ \ \mathrm{for\ any}\ x\in\mathbb{R}^{n}.$$

Again it is easy to check that $\Lambda_{\infty}K\in\mathscr{K}_{e}^{n}$. Denote by $\Lambda_{\infty}^{\circ}K$ the polar body of $\Lambda_{\infty}K$. It follows from (2.12) that

$\displaystyle\|x\|_{\Lambda_{\infty}^{\circ}K}=h_{\Lambda_{\infty}K}(x)=\max_{y\in K}\,[x,y],\ \ \ \mathrm{for\ any}\ x\in\mathbb{R}^{n}.$

(3.4)

Formula (2.8) for the Minkowski functional yields that, for any star body $L\in\mathcal{S}^{n}$, one must have $L=\big{\{}x\in\mathbb{R}^{n}:\ \|x\|_{L}\leq 1\big{\}}.$ Together with (3.4), one gets, if $K\in\mathcal{S}^{n}$,

$$\Lambda_{\infty}^{\circ}K=\Big{\{}x\in\mathbb{R}^{n}:\ \|x\|_{\Lambda_{\infty}^{\circ}K}\leq 1\Big{\}}=\Big{\{}x\in\mathbb{R}^{n}:\ \max_{y\in K}\,[x,y]\leq 1\Big{\}}.$$

This motivates our definition for the sine polar body.

Clearly, for $K\in\mathcal{S}^{n}$, one has

$$K^{\diamond}=\Lambda_{\infty}^{\circ}K.$$

(3.6)

If $K\in\mathscr{K}_{e}^{n}$, one sees that the major difference of (2.13) and (3.5) is in fact the replacement of $|x\cdot y|$ by $[x,y]$, and hence the newly defined sine polar body $K^{\diamond}$ can be viewed as the sine counterpart of $K^{\circ}$.

Throughout the paper, a closed solid cylinder $C^{-}(u,r)\subset\mathbb{R}^{n}$ with axis being the line $\{tu:t\in\mathbb{R}\}$ for $u\in S^{n-1}$ and base radius $r>0$ is a subset of $\mathbb{R}^{n}$ of the following form:

$$C^{-}(u,r)=\big{\{}x\in\mathbb{R}^{n}:[x,u]\leq r\big{\}}=\big{\{}x\in\mathbb{R}^{n}:|\textrm{P}_{u^{\perp}}x|\leq r\big{\}}.$$

Due to (2.13), one sees that if $K\in\mathscr{K}_{e}^{n}$, then $K^{\circ}$ is obtained by the intersection of slabs. However, it follows from (2.2) and (3.5) that

	$\displaystyle K^{\diamond}\!\!$	$\displaystyle=$	$\displaystyle\!\!\Big{\{}x\in\mathbb{R}^{n}:|y|\cdot|\textrm{P}_{y^{\perp}}x|\leq 1\ \ \mathrm{for\ all}\ y\in K\setminus\{o\}\Big{\}}$		(3.7)
		$\displaystyle=$	$\displaystyle\!\!\Big{\{}x\in\mathbb{R}^{n}:|\textrm{P}_{y^{\perp}}x|\leq\frac{1}{|y|}\ \ \mathrm{for\ all}\ y\in K\setminus\{o\}\Big{\}}.$

This means that $K^{\diamond}$ is indeed formed by the intersection of closed solid cylinders (with axis being the line $\{ty:t\in\mathbb{R}\}$ and base radius being $|y|^{-1}$ for $y\in K\setminus\{o\}$). As the polar body plays fundamental roles in convex geometry, such as in the celebrated Blaschke-Santaló inequality (1.1) and many other affine isoperimetric inequalities (see, e.g., [32]), we expect that the sine polar body will play roles similar to its “cosine cousin” in applications. In particular, as a start, the sine Blaschke-Santaló inequality for the sine polar body will be established in Section 5.

In view of Definition 3.1 and the nature of the sine polar body, it is worth to investigate $\mathcal{C}_{e}^{n}\subseteq\mathscr{K}_{e}^{n}$, the family of origin-symmetric convex bodies generated by the intersection of origin-symmetric closed solid cylinders. Typical examples of $\mathcal{C}_{e}^{n}$ are the Steinmetz solids, see Figure 1. As pointed out by one of the referees, our sine polarity turns out to be a special case of the notion of polarity with respect to a general function $c(\cdot,\cdot)$ introduced by Artstein-Avidan, Sadovsky, and Wyczesany [2] (namely, by letting $c(x,y)=-[x,y]$ and $t=-1$). Consequently, $\mathcal{C}_{e}^{n}$ is a type of the $c$-class of sets in [2]. These concepts were further studied in [3].

It is indeed a surprise that to generate a convex body $K\in\mathcal{C}_{e}^{n}$, as few as only two closed solid cylinders whose axes are not parallel to each other would be enough. In other words, one needs the directions of the axes of the closed solid cylinders generating $K\in\mathcal{C}_{e}^{n}$ being not concentrated on an $1$-dimensional subspace. This result is summarized in the following proposition.