The $L_p$-Petty projection inequality for $s$-concave functions

1.   Introduction

The isoperimetric inequality is a very classical problem in mathematics, which has been well known since Ancient Greece. It was first stated on the plane that the disk has the largest area among all domains enclosed by the closed curve of fixed perimeter. In the 19th century, Steiner first used the symmetrization to directly explain the existence of the solution to the isoperimetric problem, and it was not until 1870 that a rigorous and complete mathematical proof was given by Weierstrass with the variational method of analysis. Later, Hurwitz made use of the Fourier series to give a purely analytic proof. Nowadays, the isoperimetric inequality has been extended to higher-dimensional Euclidean spaces, some special manifolds, and other spaces.

Theorem 1.1. Let $C\subset \mathbb{R}^n$ be a bounded closed domain. Suppose the surface area $S$ of $C$ is fixed, then its volume $V$ is maximized when $C$ is an Euclidean ball (we call it a ball for brevity), i.e.,

with equality if and only if $C$ is a ball, where $\omega_{n}$ denotes the volume of the unit ball.

Many important mathematical ideas and methods occurred in the study of the isoperimetric inequality and have been applied to a great many mathematical disciplines. Among these, Osserman ^[15] gave its diverse implications and applications in analysis and differential geometry. Chaville ^[7] described the effects of the isoperimetric inequality in analysis. And even in physics, the isoperimetric inequality was used to study the principle of least action ^[16].

For convex bodies (compact subsets that are convex and have nonempty interiors) in $\mathbb{R}^{n}$, the isoperimetric inequality can be derived from the Brunn-Minkowski inequality. Later, Lutwak introduced the Firey $p$-Minkowski addition and generalized the classical Brunn-Minkowski theory. The following $L_{p}$ isoperimetric inequality is obtained from the $L_{p}$ Brunn-Minkowski inequality; see ^[18]. When $p = 1$, we obtain the classical isoperimetric inequality.

Theorem 1.2. Let $1\le p < \infty$ and $M$ be a convex body in $\mathbb{R}^{n}$. Then

with equality if and only if $M$ is a ball.

Another important topic is the Petty projection inequality, which is closely related to the projection body introduced by Minkowski in the last century. It indicates that the polar projection body of the ellipsoid has the largest volume among all convex bodies with fixed volume. It is important due to the fact that it is also a version of the affine isoperimetric inequality and thus further strengthens the classical isoperimetric inequality. In addition, Zhang ^[20] established the inverse inequality. Subsequently, Zhang ^[21] removed the assumption of convexity and gave the Petty projection inequality on compact sets with smooth boundaries. Soon after, Wang ^[19] generalized it to sets of finite perimeter. Moreover, Lutwak ^[13] deduced an inequality relating the volume of a convex body to the power mean of its luminosity function based on the Petty projection inequality.

In 2000, the projection body and the Petty projection inequality were generalized to the $L_p$ cases by Lutwak, Yang, and Zhang ^[12]. When $p = 1$, the following inequality is equivalent to the classical Petty projection inequality.

Theorem 1.3. Let $1\le p < \infty$ and $M$ be a convex body in $\mathbb{R}^{n}$. Then

with equality if and only if $M$ is an ellipsoid centered at the origin. Here, $\Pi_{p}^{\ast}M$ is the polar body of $L_p$-projection body $\Pi_{p}M$.

Recently, there has been a new trend towards the functionalization of geometry, which has received widespread attention in the fields of geometry and analysis. Artstein-Avidan, Klartag, and Milman ^[1] provided a connection between convex bodies and $s$-concave functions, which made the functionalization possible. For $\varphi\in {{\rm{Conc}}}_{s}(\mathbb{R}^{n})$ (see Section 2 for definitions), they defined the convex body $K_{s}(\varphi)$ in $\mathbb{R}^{n}\times \mathbb{R}^{s}$ by

where $\| \cdot \|$ denotes the Euclidean norm and $s$ is a positive integer throughout the paper. Thereafter, they further obtained a general functional form of the Blaschke-Santaló inequality based on the link above. Moreover, Milman and Rotem ^[14] studied some important inequalities on mixed integrals of $s$-concave functions. See ^{[2,4,5,6,10,17]} for the corresponding geometric inequalities on the $s$-concave functions and more research between them.

More recently, Fang and Zhou ^[9] defined the projection body $\Pi^{(s)} \varphi$ of $s$-concave functions by $\Pi^{(s)} \varphi = \Pi K_{s}(\varphi),$ for $\varphi\in {{\rm{Conc}}}_{s}(\mathbb{R}^{n})$. They also established the Petty projection inequality for $s$-concave functions.

Theorem 1.4. Let $z' = (z_{1}, \cdots, z_{n})$, $\tilde{z} = (z_{n+1}, \cdots, z_{n+s})$ and $\varphi\in {{\rm{Conc}}}_{s}^{(2)}(\mathbb{R}^{n})$. Then for all $(z', \tilde{z})\in\partial K_{s}(\varphi)$ with $z'\in {{\rm{int}}}({{\rm{supp}}}(\varphi))$,

where $M_{n, s} = (n+s)\omega_{s}\left(\frac{\omega_{n+s}}{\omega_{n+s-1}\omega_{s}}\right)^{n+s}$. Equality holds if and only if $\varphi = (a+\langle{b}, {z'}\rangle-\langle{Cz'}, {z'}\rangle)_{+}^{\frac{s}{2}}$, where $a > 0$, $b\in \mathbb{R}^n$, and $C$ is a positive definite matrix. Here $\alpha_{+} = \max\{\alpha, 0\}$ for $\alpha\in \mathbb{R}$, $\nabla \varphi$ denotes the gradient of $\varphi$, $\langle{\cdot}, {\cdot}\rangle$ denotes the inner product in Euclidean space $\mathbb{R}^{n}$, and see Section 2 for definitions of ${{\rm{Conc}}}_{s}^{(2)}(\mathbb{R}^n)$.

Besides, Fang and Zhou ^[9] established the isoperimetric inequality for $s$-concave functions.

Theorem 1.5. Let $\varphi\in {{\rm{Conc}}}_{s}^{(2)}(\mathbb{R}^{n})$. Then

where $N_{n, s} = \frac{n+s}{s}(\frac{\omega_{n+s}}{\omega_{s}})^{\frac{1}{n+s}}$. Equality holds if and only if $\varphi = (d-\left\Vert{{z'-e}}\right\Vert^{2})_{+}^{\frac{s}{2}}$, where $d > 0$ and $e\in \mathbb{R}^{n}$.

In this paper, we aim to establish analogues of Theorems 1.2 and 1.3 for $s$-concave functions. First, we give a general functional analogue of the $L_p$-projection body. Let $1\le p < \infty$ and $\varphi\in {{\rm{Conc}}}_{s}(\mathbb{R}^{n})$. We define its $L_p$-projection body $\Pi _{p}^{(s)} \varphi$ as

Note that when $p = 1$, it is the projection body $\Pi^{(s)} \varphi$ of $s$-concave functions defined in ^[9], and further, the projection body on the class of log-concave functions (the case as $s\to\infty$) has also been defined in ^[8]. Moreover, this new functional $L_p$-projection body corresponds to the $L_p$-projection body of the $n+s$ dimensional convex body associated with $\varphi$. Thus, $\Pi_{p}^{(s)}(\varphi)$ has affine invariance, continuity, and many other properties similar to the $L_p$-projection body of convex bodies.

Next, we obtain a functional analogue of the $L_p$-Petty projection inequality using analytic methods in convex geometry.

Theorem 1.6. Let $1\le p < \infty$, $z' = (z_{1}, \cdots, z_{n})$, $\tilde{z} = (z_{n+1}, \cdots, z_{n+s})$ and $\varphi\in {{\rm{Conc}}}_{s}^{(2)}(\mathbb{R}^{n})$. Then for all $(z', \tilde{z})\in\partial K_{s}(\varphi)$ with $z'\in {{\rm{int}}}({{\rm{supp}}}(\varphi))$,

where $K_{n, s} = \left[(n+s)\omega_{s}\right]^{1-\frac{n+s}{p}}\left(\frac{2}{c_{n+s-2, p}}\right)^{\frac{n+s}{p}}$. Equality holds if and only if $\varphi = (t-\langle{Dz'}, {z'}\rangle)_{+}^{\frac{s}{2}}$, where $t > 0$ and $D$ is a positive definite matrix.

In addition, we acquire the $L_p$ isoperimetric inequality for $s$-concave functions.

Theorem 1.7. Let $1\le p < \infty$, $\varphi\in {{\rm{Conc}}}_{s}^{(2)}(\mathbb{R}^{n})$ and $z' = (z_{1}, \cdots, z_{n})$. Then

where $L_{n, s} = \frac{n+s}{s}(\frac{\omega_{n+s}}{\omega_{s}})^{\frac{p}{n+s}}$. Equality holds if and only if $\varphi = (r-\left\Vert{{z'-d}}\right\Vert^{2})_{+}^{\frac{s}{2}}$, where $r > 0$ and $d\in \mathbb{R}^{n}$.

In fact, Theorems 1.6 and 1.7 are the $L_p$ extensions of Theorems 1.4 and 1.5, respectively. We remark that, although both our results and the results in ^[9] use geometric inequalities and their relation with $s$-concave functions, combining with tools in the $L_p$ Brunn-Minkowski theory for $p\geq1$, our results include the results in ^[9] as special cases when $p = 1$.

Moreover, our results generalize the geometric structures on $\mathcal K^n$ in the following sense. Let $K\in\mathcal K^n$ and $\varphi = (1-\|z'\|_K)_+^s$, $z'\in \mathbb{R}^n$. For each $t\in(0, 1]$ and $z_0\in t \mathbb{S}^{s-1}$, it is clear that

which is a dilate of $K$. In other words,

Based on Theorems 1.6 and 1.7, we make a further investigation of the connection between the $L_p$-Petty projection inequality and the $L_p$ isoperimetric inequality on the class of $s$-concave functions in Section 3.

2.   Preliminaries

In this section, we collect basics regarding convex bodies and $s$-concave functions, most of which can be found in ^[18].

Let $\mathcal{K}^n$ denote the set of all convex bodies in $\mathbb{R}^n$, and let $\mathcal{K}_{o}^n$ be the subset of $\mathcal{K}^n$ whose elements contain the origin in their interiors. Let $\partial M$ be the boundary of $M\in\mathcal{K}^n$. The unit ball and the unit sphere in $\mathbb{R}^{d}$ are denoted respectively by $B_{2}^{d} = \{z\in \mathbb{R}^{d}: \left\Vert{{z}}\right\Vert\le1\}$ and $\mathbb{S}^{d-1} = \{z\in \mathbb{R}^{d}:\left\Vert{{z}}\right\Vert = 1\}$.

For $M\in\mathcal{K}^n$, define its support function $h_{M} = h(M, \cdot): \mathbb{R}^n\to \mathbb{R}$ by

Let $M\in\mathcal{K}_{o}^n$. Define its radial function $\rho_{M} = \rho(M, \cdot): \mathbb{R}^n\setminus\{o\}\to \mathbb{R}$ by

And define the polar body $M^{\ast}$ of $M$ by

Based on the definition above, there are (see e.g., [18, Lemma 1.7.13])

For a convex body $M\in\mathcal{K}_{o}^n$, the $n$-dimensional volume of $M$ can be expressed as

In the case that $M = B_{2}^{d}$, we have $V_{d}(B_{2}^{d}) = \omega_{d} = \frac{\pi^{\frac{d}{2}}}{\Gamma(1+\frac{d}{2})}$, where $\Gamma(\cdot)$ is the Gamma function.

Let $M\in\mathcal{K}^{n}$ and $1\le p < \infty$. The support function of the $L_p$-projection body, $\Pi _{p}M$, of $M$ is defined by Lutwak, Yang, and Zhang ^[12] as

where $c_{n, p} = \frac{\omega_{n+p}}{\omega_{2}\omega_{n}\omega_{p-1}}$ and $S_{p}(M, \cdot)$ denotes the $p$-surface area measure of $M$. In particular, when $p = 1$, $S_{1}(M, \cdot)$ corresponds to the well-known surface area measure $S(M, \cdot)$ of $M$. Indeed, the measure $S_{p}(M, \cdot)$, is absolutely continuous with respect to $S(M, \cdot)$ and the Radon-Nikodym derivative is

Let $N_{M}(z)$ be the outer unit normal vector at $z\in\partial M$, which exists a.e. on $\partial M$. For $\theta = N_{M}(z)\in \mathbb{S}^{n-1}$, we have

where $\mu_{M}(\cdot)$ is the $(n-1)$-dimensional Hausdorff measure on $\partial M$. Hence, the support function $h(\Pi_{p}M, \eta)$ has the following form:

We call a function $\varphi: \mathbb{R}^n\to[0, \infty)$ $s$-concave if ${{\rm{supp}}}(\varphi) = {{\rm{cl}}}\{z\in \mathbb{R}^{n}: \varphi(z) > 0\}$ is a convex body, $\varphi$ is upper semi-continuous, and $\varphi^{\frac{1}{s}}$ is concave on ${{\rm{supp}}}(\varphi)$. Let ${{\rm{Conc}}}_{s}(\mathbb{R}^{n})$ denote the class of all $s$-concave functions, and ${{\rm{Conc}}}_{s}^{(2)}(\mathbb{R}^{n})$ denote its subset of functions that are twice continuously differentiable in the interior of their supports. It is clear that as $s$ tends to infinity, the class of $s$-concave functions converges to the class of log-concave functions in the sense of uniform convergence of compact sets, and as $s$ tends to $0$, it converges to the class of indicator functions of convex sets in the same sense as above. See ^[1,2,3] for more information.

Next, we give basics related to $K_{s}(\varphi)$. For $\varphi\in {{\rm{Conc}}}_{s}(\mathbb{R}^{n})$, by Fubini's theorem, we have

On the other hand, as a special case, setting

gives $K_{s}(\psi_{s}) = B_{2}^{n+s}$, due to the definition of $K_s(\varphi)$. As for its boundary, it again follows from the definition that

Thus, we have the following mapping:

where $z' = (z_{1}, \cdots, z_{n})\in \mathbb{R}^{n},$ and

Since $\partial K_{s}(\varphi)$ is symmetric, it suffices to consider the case that $z_{n+s} > 0$ in the calculation. Hence, the surface area element of $\partial K_{s}(\varphi)$ is given by ^[2],

Moreover, we need the following lemma.

Lemma 2.1. ^[2] Let $\varphi\in {{\rm{Conc}}}_{s}^{(2)}(\mathbb{R}^{n})$, $z' = (z_{1}, \cdots, z_{n})$ and $\tilde{z} = (z_{n+1}, \cdots, z_{n+s})$. Then for all $(z', \tilde{z})\in\partial K_{s}(\varphi)$ with $z'\in {{\rm{int}}}({{\rm{supp}}}(\varphi))$,

Here $\varphi$ is evaluated at $z'$.

3.   Main results and proofs

We first introduce the $L_p$-projection body for $s$-concave functions, using its correspondence with convex bodies.

Definition 3.1. Let $1\le p < \infty$ and $\varphi\in {{\rm{Conc}}}_{s}(\mathbb{R}^{n})$. Define the $L_p$-projection body $\Pi_{p}^{(s)} \varphi$ of $\varphi$ by

for $y\in \mathbb{R}^{n}\times \mathbb{R}^{s},$ where $z = (z_{1}, z_{2}, \cdots, z_{n+s})\in\partial K_{s}(\varphi)$ and $\lambda_{n, s, p} = (n+s)\omega_{n+s} c_{n+s-2, p}$.

Notice that $h_{\Pi_{p}^{(s)} \varphi}$ is defined by the support function of the convex body $\Pi_{p}K_s(\varphi)$; hence, it is convex. Therefore, its continuity on compact subsets of $\mathbb{R}^{n+s}$ follows (see e.g., [18, Theorem 1.5.3]).

In addition, we use its following form to facilitate later calculations.

Lemma 3.1. Let $1\le p < \infty$ and $\varphi\in {{\rm{Conc}}}_{s}^{(2)}(\mathbb{R}^{n}).$ For each $\theta\in \mathbb{S}^{n+s-1}$, we have

where $z' = (z_{1}, \cdots, z_{n})\in \mathbb{R}^{n}$ and $\tilde{z} = (z_{n+1}, \cdots, z_{n+s})\in \mathbb{R}^{s}$.

Proof. Let $\theta\in \mathbb{S}^{n+s-1}$, $z' = (z_{1}, \cdots, z_{n})$, $\tilde{z} = (z_{n+1}, \cdots, z_{n+s})$ such that $z = (z', \tilde{z})\in(\mathbb{R}^{n}\times \mathbb{R}^{s})\cap\partial K_{s}(\varphi)$. And denote $\tilde{\partial}K_{s}(\varphi) = \Big\{z\in\partial K_{s}(\varphi):z'\in {{\rm{int}}}({{\rm{supp}}}(\varphi))\Big\}.$ Since $\partial K_{s}(\varphi)\setminus\tilde{\partial} K_{s}(\varphi)$ has measure zero, by Definition 3.1, Lemma 2.1, and (2.3), we obtain

The equation above follows from the symmetry of $\partial K_{s}(\varphi)$. Here $\varphi$ is evaluated at $z' = (z_{1}, \cdots, z_{n})\in \mathbb{R}^{n}$. □

Next, in light of Lemma 3.1 and the $L_p$-Petty projection inequality on convex bodies, we give the proof of Theorem 1.6.

Proof of Theorem 1.6. Let $\Pi_{p}^{(s), \ast}(\varphi)$ be the polar body of $\Pi _{p}^{(s)}(\varphi)$. By Lemma 3.1, Theorem 1.3, and (2.2), we obtain

where $\alpha_{n, s, p} = \frac{1}{n+s}\left(\frac{\lambda_{n, s, p}}{2} \right)^{\frac{n+s}{p}}.$ This gives the desired $L_p$-Petty projection inequality for $s$-concave functions.

From the proof above, it is clear that the equality holds if and only if $K_{s}(\varphi)$ is an ellipsoid with the origin as its center due to Theorem 1.3. It is further equivalent to $\varphi = (t-\langle Dz', z'\rangle)_{+}^{\frac{s}{2}}$, where $t > 0$ and $D$ is a positive definite matrix, since $K_s(\varphi)$ clearly corresponds to an ellipsoid by the definition in this case. □

Now, we prove the $L_p$ isoperimetric inequality on the class of $s$-concave functions.

Proof of Theorem 1.7. Using (2.1) and the similar treatment as in the proof of Lemma 3.1, we obtain

Applying (3.3) in ^[9], we have

and then

Therefore, Theorem 1.2 and (2.2) imply

and Theorem 1.7 follows. The equality holds if and only if $K_{s}(\varphi)$ is a ball, which is equivalent to $\varphi = (r-\|z'-d\|^{2})_{+}^{\frac{s}{2}}$, where $r > 0$ and $d\in \mathbb{R}^n$. □

On convex bodies, Lutwak ^[11] pointed out that the classical isoperimetric inequality can be deduced from the Petty projection inequality. After our verification, we found a corresponding conclusion on $s$-concave functions, as shown in the next result.

Theorem 3.1. The $L_p$-Petty projection inequality for $s$-concave functions

strengthens the $L_p$ isoperimetric inequality for $s$-concave functions

Proof. First, notice that $\|(\varphi^{\frac1s}\nabla \varphi^{\frac1s}, -\tilde{z})\| = \varphi^{\frac1s}(1+\|\nabla \varphi^{\frac1s}\|^{2})^{\frac12}$, and that, for $y\in \mathbb{R}^{n+s}$, $\eta = \frac{y}{\|y\|}$,

Next, we use Jensen's inequality, Fubini's theorem, and the results above to obtain

It follows that

where $\beta_{n, s} = \frac{1}{(n+s)\omega_{n+s}}.$ Therefore, we obtain the desired result. □

4.   Conclusions

This work gives a clear picture of the correspondence between concepts and inequalities in convex geometry and those in analysis on $s$-concave functions. It has important applications in integral geometry, differential geometry, image analysis, and so on. It is among the leading focuses in the field of convex geometry. Based on the mapping $K_s$ that maps from an $s$-concave function to a convex body, defined by Artstein-Avidan, Klartag, and Milman, and the work by Fang and Zhou in the case when $p = 1$, in this work, on the class of $s$-concave functions, a general function counterpart of the $L_p$-projection body is introduced. In addition, the $L_p$-Petty projection inequality and the $L_p$ isoperimetric inequality are established on this class. Finally, we show that the $L_p$-Petty projection inequality strengthens the $L_p$ isoperimetric inequality on the same class. We believe this work serves as an important bridge between finite-dimensional convex geometry and infinite-dimensional functional analysis and thus contributes to the advances of the functionalization of convex geometry.

Author contributions

Tian Gao and Dan Ma: Conceptualization, investigation, writing. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors wish to thank the referees for valuable suggestions and careful reading of the original manuscript. The work of the authors is supported in part by the National Natural Science Foundation of China (Grant No. 12471055) and the Natural Science Foundation of Gansu Province (Grant No. 23JRRG0001).

Conflict of interest

The authors declare that they have no competing interests.