1 Introduction

Throughout this paper, a convex body is a nonempty convex compact set in the n-dimensional Euclidean space \(\mathbb {R}^n\). We denote the set of convex bodies in \(\mathbb {R}^n\) by \(\mathcal {K}^n\), and the subset of convex bodies in \(\mathbb {R}^n\) containing the origin o in their interiors by \(\mathcal {K}_0^n\). We use \(\langle x,y \rangle \) to denote the inner product of the vectors \(x,y\in \mathbb {R}^n\), and \(\Vert x\Vert \) to denote the Euclidean norm of x. For \(K\in \mathcal {K}^n\), the support function \(h_K: \mathbb {R}^n\rightarrow \mathbb {R}\) of K is defined for \(x\in \mathbb {R}^n\) by

$$\begin{aligned} h_K(x)=\max \{\langle x,y\rangle : y\in K\}. \end{aligned}$$
(1.1)

If \(K\in \mathcal {K}_0^n\), then the polar body \(K^{\circ }\) of K is defined by \(K^{\circ }=\{x\in \mathbb {R}^n: \langle x,y\rangle \le 1 \ \text {for all} \ y\in K\}.\) The unit sphere \(\{x\in \mathbb {R}^n:\Vert x\Vert =1\}\) is denoted by \(S^{n-1}\) and the unit ball \(\{x\in \mathbb {R}^n:\Vert x\Vert \le 1\}\) by \(B_2^n\). Furthermore, we use \(|\cdot |\) to denote the volume of a convex body in the appropriate dimension. In particular, we write \(\kappa _{n}=|B_2^n|=\frac{\pi ^{n/2}}{\Gamma (\frac{n}{2}+1)}\), where \(\Gamma (\cdot )\) represents the Gamma function.

The classical Steiner formula states that for \(K\in \mathcal {K}^n\) and the unit ball \(B^{n}_{2}\) in \(\mathbb {R}^n\), the volume of the Minkowski sum \(K+t B^{n}_{2}\) can be expressed as a polynomial of degree n in the parameter \(t\ge 0\). More precisely,

$$\begin{aligned} |K+t B^{n}_{2}|=\sum _{i=0}^n \kappa _{n-i}V_{i}(K)t^{n-i}, \end{aligned}$$

where \(K+t B^{n}_{2}=\{x+t y: x\in K, y\in B_2^n\}\) and \(V_{i}(K)\) is the i-th intrinsic volume of K, \(0\le i\le n\). The notion of intrinsic volume introduced by McMullen [14] extends the usual concept of volume, i.e., \(V_n(K)=|K|\). If K is j-dimensional, then \(V_j(K)\) is exactly the j-dimensional volume of K, which means that intrinsic volumes do not depend on the dimension of the ambient space. Moreover, up to constants, \(V_{n-1}(K)\) and \(V_{1}(K)\) are the surface area and the mean width of K, respectively. It is worth noting that up to constants and reindexing, the intrinsic volumes are also called quermassintegrals. Quermassintegrals are a special case of the general mixed volumes, which are the main objects studied in the classical Brunn–Minkowski theory developed by Minkowski, Aleksandrov, Fenchel, Blaschke, and others in the earlier part of the twentieth century. For further details, see Schneider’s book [16].

To study the relation with the lattice-point enumerator, Wills [21] introduced the following functional

$$\begin{aligned} \mathcal {W}(K)=\sum _{i=0}^nV_{i}(K), \end{aligned}$$

for \(K\in \mathcal {K}^n\). This functional is now called the Wills functional, which has many interesting properties. For instance, Hadwiger [7] showed that \(\mathcal {W}(K)\) can be expressed by several elegant integral representations:

$$\begin{aligned} \mathcal {W}(K)=\int _{\mathbb {R}^n}e^{-\pi d(x,K)^2}dx=2\pi \int _{0}^{\infty } |K+tB_{2}^{n}|te^{-\pi t^2}\textrm{d}t, \end{aligned}$$
(1.2)

where \(d(x,K)=\min _{y\in K}\Vert x-y\Vert \) is the Euclidean distance from x to K. Hadwiger also showed in [7] that for any subspace H of \(\mathbb {R}^n\) and two convex bodies \(K\subset H\) and \(L\subset H^\bot \), where \( H^{\perp }\) is the orthogonal complement of H,

$$\begin{aligned} \mathcal {W}(K+L)=\mathcal {W}(K)\mathcal {W}(L). \end{aligned}$$
(1.3)

An upper bound of \(\mathcal {W}(K)\) in terms of \(V_1(K )\) was established by McMullen [15]:

$$\begin{aligned} \mathcal {W}(K)\le e^{V_1(K)}. \end{aligned}$$
(1.4)

Meanwhile, this functional has found numerous applications in different areas of mathematics, such as discrete geometry [23], probability theory [17,18,19,20], and complex analysis [9, 10, 22].

Recently, Alonso–Gutiérrez, Hernández Cifre, and Yepes Nicolás [2] reproved McMullen’s upper bound and provided a new lower bound for the Wills functional. More specifically, they showed that for \(K\in \mathcal {K}^n\),

$$\begin{aligned} e^{V_{1}(K)-\pi c(K)^{2}} \le \mathcal {W}(K) \le e^{V_{1}(K)}. \end{aligned}$$
(1.5)

where \(c(K)= \min \{t>0: K \subset x+tB_{2}^{n}, \ x\in \mathbb {R}^n\}\) is the circumradius of K.

In the 1970s, the dual Brunn–Minkowski theory was initiated by Lutwak [12], which presents a highly nontrivial duality in convex geometry by replacing the Minkowski sum with the radial sum. This theory has rapidly expanded over the last forty years. For a detailed bibliography on these topics, we refer the reader to [16, Chapter 9] and the references therein. If K and L are two compact star-shaped sets with respect to the origin in \(\mathbb {R}^n\) and \(a,b\ge 0\), then the radial sum \(aK\widetilde{+}bL\) is a compact star-shaped set defined by

$$\begin{aligned} aK\widetilde{+}bL=\{ax+by: x \in K, y\in L \ \textrm{and} \ \langle x,y\rangle =\Vert x\Vert \Vert y\Vert \}. \end{aligned}$$

The condition \(\langle x,y\rangle =\Vert x\Vert \Vert y\Vert \) means that x and y are collinear with the origin. For a compact star-shaped set \(K\subset \mathbb {R}^n\), its radial function \(\rho _K:\mathbb R^n\backslash \{o\}\rightarrow \mathbb R\) is defined by \(\rho _K(x)=\max \{\lambda \ge 0:\lambda x\in K\}.\) If \(\rho _K\) is positive and continuous, then we call K a star body.

The dual Steiner formula introduced by Lutwak [13] states that, for a star body K in \(\mathbb {R}^n\),

$$\begin{aligned} |K\widetilde{+}t B^{n}_{2}|=\sum _{i=0}^n\kappa _{n-i} \widetilde{V}_{i}(K)t^{n-i}, \quad t\ge 0, \end{aligned}$$
(1.6)

where \(\widetilde{V}_{i}(K)\) is the i-th dual volume of K. Here, we use the normalization \(\widetilde{V}_{i}(B_2^n)=V_{i}(B_2^n)\) for \(\widetilde{V}_{i}\). The i-th dual volume has the integral representation

$$\begin{aligned} \widetilde{V}_{i}(K)= \left( {\begin{array}{c}n\\ i\end{array}}\right) \frac{1}{n\kappa _{n-i}}\int _{S^{n-1}}\rho _{K}(u)^{i}\textrm{d}u,\quad 0\le i\le n, \end{aligned}$$

where du denotes the usual spherical Lebesgue measure. In particular, \(\widetilde{V}_{0}(K)=1\) and \(\widetilde{V}_{n}(K)=|K|\).

Thus, the dual Wills functional \(\widetilde{\mathcal {W}}(K)\) was naturally introduced by Besau, Hoehner, and Kur [5]. For \(K\in \mathcal {K}_0^n\),

$$\begin{aligned} \widetilde{\mathcal {W}}(K)=\sum _{i=0}^n\widetilde{V}_{i}(K), \end{aligned}$$

which can be represented as

$$\begin{aligned} \widetilde{\mathcal {W}}(K)=\int _{\mathbb {R}^n}e^{-\pi \widetilde{d}(x,K)^2}\textrm{d}x, \end{aligned}$$
(1.7)

where \(\widetilde{d}(x,K)\) is the minimal radial distance of x to K, or equivalently,

$$\begin{aligned} \widetilde{d}(x,K)={\left\{ \begin{array}{ll} ||x||-\rho _{K}(x/||x||),\quad x\notin K,\\ 0,\qquad \qquad \qquad \qquad \qquad x\in K. \end{array}\right. } \end{aligned}$$
(1.8)

In other words,

$$\begin{aligned} \widetilde{d}(x,K)=\min _{y \in K\cap l_{x}}\Vert x-y\Vert , \end{aligned}$$
(1.9)

where \(l_x\) is the line through the origin and x.

In this paper, we aim to explore some properties of the dual Wills functional, which are the part of the dual Brunn–Minkowski theory. Basic properties for the dual Wills functional can be listed as follows.

  1. (i)

    For a star body K in \(\mathbb {R}^n\), there is an alternative integral expression for the dual Wills functional.

    $$\begin{aligned} \widetilde{\mathcal {W}}(K)=\int _{\mathbb {R}^n}e^{-\pi \widetilde{d}(x,K)^2}\textrm{d}x=2\pi \int _{0}^{\infty } |K\widetilde{+}tB_{2}^{n}|te^{-\pi t^2}\textrm{d}t. \end{aligned}$$
    (1.10)
  2. (ii)

    For \(K\in \mathcal {K}_0^n\), we have

    $$\begin{aligned} \widetilde{\mathcal {W}}(K)\le \mathcal {W}(K). \end{aligned}$$

    For \(n=1\), equality holds if and only if K is a closed interval in \(\mathbb {R}\) containing the origin in its interior; for \(n\ge 2\), equality holds if and only if K is a ball centered at the origin.

  3. (iii)

    Let \(G_{n,k}\) denote the Grassmann manifold of all k-dimensional linear subspaces of \(\mathbb {R}^n\). For \(H\in G_{n,k}\), if two star bodies \(K_1\subset H\) and \(K_2\subset H^\bot \) contain the origin in their relative interiors, then

    $$\begin{aligned} \widetilde{\mathcal {W}}(K_1+ K_2) \le \widetilde{\mathcal {W}}(K_1)\widetilde{\mathcal {W}}(K_2). \end{aligned}$$

Property (i) can be found in [5]. Property (ii) is a direct consequence of Corollary 1.4 of [12]. Property (iii) will be proved in Sect. 2.

For a star body K in \(\mathbb {R}^n\), let \(R(K)= \min \{t>0: K \subset tB_{2}^{n}\}\) be the outer radius of K and let \(r(K)= \max \{t>0: K \supset tB_{2}^{n}\}\) be the inner radius of K. Denote by \(H_K\) the hook function of \(\big (\frac{R(K)}{r(K)}\big )^{\frac{n}{2}}\), i.e., \(H_K=\frac{1}{2}\Big (\big (\frac{R(K)}{r(K)}\big )^{\frac{n}{2}}+\big (\frac{r(K)}{R(K)}\big )^{\frac{n}{2}}\Big )\). Following the idea of Alonso–Gutiérrez et al. [2], the upper and lower bounds for the dual Wills functional in terms of \(\widetilde{V_{1}}\) are obtained as follows (see Theorems 2.2 and 2.5).

Theorem 1.1

If K is a star body in \(\mathbb {R}^n\), then

$$\begin{aligned} e^{-\pi R(K)^{2}}e^{\widetilde{V_{1}}(K)}\le \widetilde{\mathcal {W}}(K)\le H_K^{2} e^{\widetilde{V_{1}}(K)}. \end{aligned}$$

A convex body \(K\in \mathcal {K}_0^n\) is said to be in John position if \(B_{2}^{n}\) is the unique maximal volume ellipsoid contained in K. The well-known John theorem [11] states that K is in John position if and only if for some \(m\ge n\) there are unit vectors \((u_i)_1^m\) on the boundary of K and positive numbers \((c_i)_1^m\) satisfying \(\sum _{i=1}^mc_iu_i=0\) and

$$\begin{aligned} \sum _{i=1}^mc_iu_i\otimes u_i=I_n, \end{aligned}$$
(1.11)

where \(u_i\otimes u_i\) is the rank-one orthogonal projection onto the space spanned by \(u_i\) and \(I_n\) is the identity map on \(\mathbb {R}^n\). It was proved by Alonso–Gutiérrez and Brazitikos [1] that for any \(H\in G_{n,k}\) and for a centrally symmetric convex body \(K\in \mathcal {K}_0^n\) in John position,

$$\begin{aligned} \mathcal {W}(\lambda (K\cap H)) \le \mathcal {W}\Big (\lambda \sqrt{\frac{n}{k}}B^{k}_{\infty }\Big )=\Big (1+2\lambda \sqrt{\frac{n}{k}}\Big )^k,\quad \lambda \ge 0, \end{aligned}$$
(1.12)

where \(B_{\infty }^n\) is the unit ball of the \(\ell _{\infty }^n\)-space.

The concept of John position can be generalized to a more general situation. In particular, a finite Borel measure \(\mu \) on \(S^{n-1}\) is said to be isotropic if

$$\begin{aligned} \int _{S^{n-1}}u\otimes ud\mu (u)=I_n. \end{aligned}$$
(1.13)

Note that it is impossible for an isotropic measure to be concentrated on a proper subspace of \(\mathbb {R}^n\). The measure \(\mu \) is said to be even if it assumes the same value on antipodal sets. Condition (1.13) reduces to (1.11) if the isotropic measure \(\mu \) is of the form \(\sum _{i=1}^mc_i\delta _{u_i}\) or \((1/2)\sum _{i=1}^m(c_i\delta _{u_i}+c_i\delta _{-u_i})\) on \(S^{n-1}\) (\(\delta _{x}\) stands for the Dirac mass at x). If \(\mu \) is an even isotropic measure on \(S^{n-1}\), then we can define a convex body C in \(\mathbb {R}^n\) as the convex hull of the support set of \(\mu \) (\(\textrm{supp}\,\mu \)), i.e., \(C=\text {conv}\{\text {supp}\,\mu \}.\) Clearly, the convex body C must contain the origin in its interior. Then, the polar body \(C^{\circ }\)of C is given by

$$\begin{aligned} C^{\circ }=\{x\in \mathbb {R}^n: \langle x,u\rangle \le 1 \ \text {for all} \ u\in \text {supp}\,\mu \}. \end{aligned}$$
(1.14)

Obviously, C and \(C^{\circ }\) are both origin symmetric, and \(C\subseteq B_2^n\) and \(B_2^n\subseteq C^{\circ }\).

Motivated by the work of Alonso–Gutiérrez and Brazitikos [1], we establish the following inequality by using the continuous version of the Brascamp–Lieb inequality due to Barthe [4] (or see Theorem 3.1).

Theorem 1.2

Let \(\mu \) be an even isotropic measure on \(S^{n-1}\) and \(C=\textrm{conv}\{\textrm{supp}\,\mu \}\). For any \(H\in G_{n,k}\), we have

$$\begin{aligned} \widetilde{\mathcal {W}}(\lambda (C^\circ \cap H))\le \mathcal {W}\Big (\lambda \sqrt{\frac{n}{k}}B^{k}_{\infty }\Big ), \quad \lambda \ge 0. \end{aligned}$$

The paper is organized as follows. We will give the upper and lower bounds of the dual Wills functional for star bodies in Sect. 2. Theorem 1.2 will be proved in Sect. 3.

2 The Upper and Lower Bounds of the Dual Wills Functional

We now prove Property (iii) as follows.

Lemma 2.1

For \(H\in G_{n,k}\), if two star bodies \(K_1\subset H\) and \(K_2\subset H^\bot \) contain the origin in their relative interiors, then

$$\begin{aligned} \widetilde{\mathcal {W}}(K_1+ K_2) \le \widetilde{\mathcal {W}}(K_1)\widetilde{\mathcal {W}}(K_2). \end{aligned}$$
(2.1)

Proof

Let \(K=K_1+ K_2\), which is obviously a star body in \(\mathbb {R}^n\) containing the origin in its interior. For \(x\notin K\), let \(x_0\in K\) be such that

$$\begin{aligned} \widetilde{d}(x,K)=||x-x_{0}||. \end{aligned}$$

There is

$$\begin{aligned} \widetilde{d}(x,K)^2&=||x-x_{0}||^2\\ {}&=\Vert \textrm{P}_{H}x-\textrm{P}_{H}x_{0}\Vert ^2+\Vert \textrm{P}_{H^\perp }x-\textrm{P}_{H^\perp }x_{0}\Vert ^2\\&\ge \widetilde{d}(\textrm{P}_{H}x,\textrm{P}_{H}K)^{2}+\widetilde{d}(\textrm{P}_{H^\perp }x,\textrm{P}_{H^\perp }K)^{2}, \end{aligned}$$

where \(\textrm{P}_{H}\) and \(\textrm{P}_{H^\perp }\) denote the orthogonal projections into H and \(H^\perp \), respectively. And if \(x\in K\), then \(\widetilde{d}(x,K)=\widetilde{d}(\textrm{P}_{H}x,\textrm{P}_{H}K) =\widetilde{d}(\textrm{P}_{H^\perp }x,\textrm{P}_{H^\perp }K)=0\). So we have

$$\begin{aligned} \widetilde{\mathcal {W}}(K_1+ K_2)&=\int _{\mathbb {R}^n}e^{-\pi \widetilde{d}(x,K)^{2}}\textrm{d}x\\ {}&\le \int _{\mathbb {R}^n}e^{-\pi \widetilde{d}(\textrm{P}_{H}x,K_{1})^{2}-\pi \widetilde{d}(\textrm{P}_{H^\perp }x,K_{2})^{2}}\textrm{d}x\\&=\int _{H}e^{-\pi \widetilde{d}(x_1,K_{1})^{2}}\textrm{d}x_1\int _{H^\perp }e^{-\pi \widetilde{d}(x_2,K_{2})^{2}}\textrm{d}x_2\\&=\widetilde{\mathcal {W}}(K_1)\widetilde{\mathcal {W}}(K_2). \end{aligned}$$

\(\square \)

An analogous proof to the one of Theorem 1.2 in [2] leads to the following result.

Theorem 2.2

If K is a star body in \(\mathbb {R}^n\), then

$$\begin{aligned} \widetilde{\mathcal {W}}(K)\ge e^{\widetilde{V_{1}}(K)-\pi R(K)^{2}}. \end{aligned}$$

To establish the upper bound of the dual Wills functional, we will introduce some notations. A function\( f: \mathbb R^n \rightarrow [0,+\infty )\) is said to be radial log-concave if it is of the form \(f=e^{-g}\) with \(g: \mathbb R^n \rightarrow (-\infty ,+\infty ]\) being a radial convex function, i.e., for \(\lambda \in (0,1)\) and \(y \in l_{x}\),

$$\begin{aligned} g(\lambda x+(1-\lambda )y)\le \lambda g(x)+(1-\lambda )g(y). \end{aligned}$$

Obviously, if f(x) is a log-concave (or a convex) function, then it must be a radial log-concave (a radial convex) function. Other related concepts (e.g., dual quasi-concave functions) can be found in [8]. If \(f:\mathbb {R}^{n} \rightarrow (-\infty ,\infty ]\) is radial convex, then the radial Legendre transformation \(\mathcal {R} f:\mathbb {R}^{n} \rightarrow \mathbb {R}\) can be defined by

$$\begin{aligned} \mathcal {R}f(x)=\sup \limits _{y \in l_{x}}(\langle x,y\rangle -f(y)). \end{aligned}$$
(2.2)

For a radial log-concave function \(f: \mathbb R^n \rightarrow [0,+\infty )\), its radial polar function \(f^\bullet \) is defined by

$$\begin{aligned} f^{\bullet }=\mathop {\inf }\limits _{y\in l_{x}}\frac{e^{-\langle x,y \rangle }}{f(y)}. \end{aligned}$$

It is easy to see that

$$\begin{aligned} -\log f^{\bullet }=\mathcal {R}(-\log f). \end{aligned}$$
(2.3)

For a star body K in \(\mathbb {R}^n\), we define the radial polar body \(K^\star \) of K by

$$\begin{aligned} K^{\star }=\{x\in \mathbb {R}^n:\langle x,y \rangle \le 1\quad \text {for all}~y\in K \ \textrm{and} \ x\in l_{y}\}. \end{aligned}$$

It immediately follows that for any \(u\in S^{n-1}\),

$$\begin{aligned} \rho _K(u)\rho _{K^\star } (u)=1. \end{aligned}$$
(2.4)

Lemma 2.3

If K is a star body in \(\mathbb {R}^n\), then

$$\begin{aligned} |K||K^{\star }|\le H_K^2\kappa _{n}^2. \end{aligned}$$
(2.5)

To prove Lemma 2.3, we will use the following inverse Cauchy-Schwarz inequality (see, e.g., [6]): If \((X,\Sigma ,\mu )\) is a positive measure space, and \(f_1, f_2 \in L^{2}(X,\Sigma ,\mu )\) satisfying

$$\begin{aligned} 0<m_{1}\le f_1\le M_{1}<\infty ,\quad 0<m_{2}\le f_2\le M_{2} <\infty , \end{aligned}$$

then

$$\begin{aligned} \int _{X} f_1^{2}\textrm{d}\mu \int _{X} f_2^{2}\textrm{d}\mu \le \frac{(m_{1}m_{2}+M_{1}M_{2})^{2}}{4m_{1}m_{2}M_{1}M_{2}} \Big (\int _{X}f_1f_2\textrm{d}\mu \Big )^{2}. \end{aligned}$$
(2.6)

Proof

(Proof of Lemma 2.3) For a star body K in \(\mathbb {R}^n\), it is easy to see that

$$\begin{aligned} \max _{u\in S^{n-1}}\rho _{K}(u)=R(K)=\frac{1}{r(K^\star )} \end{aligned}$$

and

$$\begin{aligned} \min _{u\in S^{n-1}}\rho _{K}(u)=r(K)=\frac{1}{R(K^\star )}. \end{aligned}$$

Thus, (2.4) and (2.6) yield that

$$\begin{aligned} H_K\kappa _{n}&=\frac{H_K}{n}\int _{S^{n-1}}(\rho _K(u)\rho _{K^{\star }}(u))^{\frac{n}{2}} \textrm{d}u\nonumber \\ {}&\ge \Big (\frac{1}{n}\int _{S^{n-1}}\rho _K^{n}(u)\textrm{d}u\Big )^{\frac{1}{2}}\Big (\frac{1}{n}\int _{S^{n-1}}\rho _{K^{\star }}^{n}(u)\textrm{d}u\Big )^{\frac{1}{2}}\nonumber \\&=(|K||K^{\star }|)^{\frac{1}{2}}. \end{aligned}$$

\(\square \)

The proof of Lemma 2.4 is just the same as the one of Lemma 3.1 in [2]. So we omit the details.

Lemma 2.4

Let K be a star body in \(\mathbb {R}^n\). Then for \(x\in \mathbb {R}^n\),

$$\begin{aligned} \mathcal {R}(\pi \widetilde{d}(\cdot ,K)^{2})(x)=\frac{\Vert x\Vert ^{2}}{4\pi }+\Vert x\Vert \rho _{K}(x/\Vert x\Vert ). \end{aligned}$$

Theorem 2.5

If K is a star body in \(\mathbb {R}^n\), then

$$\begin{aligned} \widetilde{\mathcal {W}}(K)\le H_K^2e^{\widetilde{V_{1}}(K)}. \end{aligned}$$

Proof

By (2.3), Lemma 2.4, making the change of variable \(x=\sqrt{2\pi }z\), Jensen’s inequality, and using polar coordinates, we have

$$\begin{aligned} \int _{\mathbb {R}^n}\big (e^{-\pi \widetilde{d}(\cdot ,K)^{2}}\big )^{\bullet }(x)\textrm{d}x&=\int _{\mathbb {R}^n}e^{-\mathcal {R}(\pi \widetilde{d}(\cdot ,K)^{2})(x)}\textrm{d}x\nonumber \\&=\int _{\mathbb {R}^n}e^{-(\frac{\Vert x\Vert ^{2}}{4\pi }+\Vert x\Vert \rho _{K}(x/\Vert x\Vert ))}\textrm{d}x\nonumber \\&=(2\pi )^{n/2}\int _{\mathbb {R}^n}e^{-(\frac{\Vert z\Vert ^{2}}{2} +(2\pi )^{1/2}\Vert z\Vert \rho _{K}(z/\Vert z\Vert ))}\textrm{d}z\nonumber \\&=(2\pi )^{n}\int _{\mathbb {R}^n}e^{-((2\pi )^{1/2}\Vert z\Vert \rho _{K}(z/\Vert z\Vert ))}\textrm{d}\gamma (z)\nonumber \\&\ge (2\pi )^{n}e^{-((2\pi )^{1/2}\int _{\mathbb {R}^n}\Vert z\Vert \rho _{K}(z/\Vert z\Vert )\textrm{d}\gamma (z))}\nonumber \\&=(2\pi )^{n}\exp \Big \{-(2\pi )^{\frac{1-n}{2}}\int _{S^{n-1}}\int ^{\infty }_{0}e^{-\frac{r^{2}}{2}}\rho _{K}(u)r^{n}\textrm{d}r\textrm{d}u\Big \}\nonumber \\&=(2\pi )^{n}\exp \Big \{-(2\pi )^{\frac{1-n}{2}}\int _{S^{n-1}}\rho _{K}(u)\Big (\int ^{\infty }_{0}r^{n}e^{-\frac{r^2}{2}}\textrm{d}r\Big )\textrm{d}u\Big \}\nonumber \\&=(2\pi )^{n}e^{-\widetilde{V_{1}}(K)}. \end{aligned}$$
(2.7)

Denote the level set of a function f on \(\mathbb {R}^n\) by

$$\begin{aligned} K_{f}(t)=\{x\in \mathbb R^n: f(x)\le t\} \end{aligned}$$

for any \(t>0\). An analogous proof to the idea of Ball in [3] leads to the following inclusion: for \(s,t>0\),

$$\begin{aligned} K_{\mathcal {R}f}(s)\subset (s+t)(K_{f}(t))^{\star }. \end{aligned}$$
(2.8)

In fact, if \(x\in K_{f}(t)\), \(y\in K_{\mathcal {R}f}(s)\) and \(y\in l_{x}\), we have \(\langle x,y \rangle \le f(x)+\mathcal {R}f(y)\le s+t\), which implies that \(\langle x,\frac{y}{s+t} \rangle \le 1\). Thus, \(\frac{y}{s+t}\in (K_{f}(t))^{\star }\), and the inclusion follows.

Let \(f(\cdot )=\pi \widetilde{d}(\cdot ,K)^{2}\). Then, for \(t\>0\),

$$\begin{aligned} K_{f}(t)=\bigg \{x\in \mathbb {R}^n: \widetilde{d}(x,K)\le \sqrt{\frac{t}{\pi }}\bigg \} =K\widetilde{+}\sqrt{\frac{t}{\pi }}B_{2}^{n}, \end{aligned}$$

and \(\frac{R(K_{f}(t))}{r(K_{f}(t))}=\frac{R(K)+\sqrt{\frac{t}{\pi }}}{r(K)+\sqrt{\frac{t}{\pi }}}\). It is easy to check that

$$\begin{aligned} 1\le \frac{R(K)+\sqrt{\frac{t}{\pi }}}{r(K)+\sqrt{\frac{t}{\pi }}}\le \frac{R(K)}{r(K)}, \end{aligned}$$

and the monotonic increase of the hook function on \([1,\infty )\) implies \(H_{K_{f}(t)}\le H_{K}\) for any \(t>0\). Now, by Lemma 2.3 and (2.8), we have

$$\begin{aligned} e^{-s}|K_{\mathcal {R}f}(s)|\cdot e^{-t}|K_{f}(t)|\le H_{K_{f}(t)}^2e^{-(s+t)}(t+s)^{n}\kappa _{n}^2\le H_{K}^2e^{-(s+t)}(t+s)^{n}\kappa _{n}^2. \end{aligned}$$

The 1-dimensional Prékopa–Leindler inequality states that if \(h_1,h_2,h_3:\mathbb {R} \rightarrow [0,+\infty )\) are integrable functions such that, for \(\lambda \in (0,1)\) and any \(x,y \in \mathbb {R}\), one has \(h_3((1-\lambda )x+\lambda y)\ge h_1(x)^{1-\lambda }h_2(y)^{\lambda }\), then

$$\begin{aligned} \int _{\mathbb {R}}h_3(x)\textrm{d}x\ge \Big (\int _{\mathbb {R}}h_1(x)\textrm{d}x\Big )^{1-\lambda } \Big (\int _{\mathbb {R}}h_2(x)\textrm{d}x\Big )^{\lambda }. \end{aligned}$$
(2.9)

So let \(h_{1}(s)=e^{-s}|K_{\mathcal {R}f}(s)|\), \(h_{2}(t)=e^{-t}|K_{f}(t)|\), \(h_{3}(t)=H_{K}e^{-t}(2t)^{\frac{n}{2}}\kappa _{n}\). Then, by (2.7), Fubini’s theorem, and the Prékopa–Leindler inequality (2.9) with \(\lambda =\frac{1}{2}\), we obtain

$$\begin{aligned} \widetilde{\mathcal {W}}(K)(2\pi )^{n}e^{-\widetilde{V_{1}}(K)}&\le \widetilde{\mathcal {W}}(K)\int _{\mathbb {R}^n}(e^{-\pi \widetilde{d}(\cdot ,K)^{2}})^{\bullet }(x)\textrm{d}x\\&=\int _{\mathbb {R}^n}e^{-\pi \widetilde{d}(x,K)^{2}}\textrm{d}x\int _{\mathbb {R}^n}e^{-\mathcal {R}(\pi \widetilde{d}(\cdot ,K)^{2})(x)}\textrm{d}x\\&=\int _{0}^{\infty }e^{-t}|K_{f}(t)|\textrm{d}t\int _{0}^{\infty }e^{-t}|K_{\mathcal {R}f}(t)|\textrm{d}t\\&\le \bigg (\int _{0}^{\infty } H_{K}e^{-t}(2t)^{\frac{n}{2}}\kappa _{n}\textrm{d}t\bigg )^{2}\\ {}&=H_{K}^2(2\pi )^{n}, \end{aligned}$$

which gives the desired result. \(\square \)

3 An Inequality of Sections of Convex Bodies for Isotropic Measures

Recall that a Borel measure \(\mu \) on \(S^{n-1}\) is isotropic provided

$$\begin{aligned} I_n=\int _{S^{n-1}}u\otimes u\textrm{d}\mu (u). \end{aligned}$$
(3.1)

Taking the trace on both sides of (3.1) yields

$$\begin{aligned} \mu (S^{n-1})=n. \end{aligned}$$
(3.2)

For \(H\in G_{n,k}\) and any Borel set \(A\subset S^{n-1}\cap H\), we define the Borel measure \(\bar{\mu }\) on \(S^{n-1}\cap H\) by

$$\begin{aligned} \bar{\mu }(A)=\int _{S^{n-1}\setminus H^{\perp }}{} {\textbf {1}}_A\Big (\frac{\text {P}_Hu}{\Vert \text {P}_Hu\Vert }\Big )\Vert \text {P}_Hu\Vert ^2\textrm{d}\mu (u). \end{aligned}$$
(3.3)

where

$$\begin{aligned} {\textbf {1}}_{A}(x)={\left\{ \begin{array}{ll} 1,\quad x\in A\\ 0,\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

For an arbitrary \(y\in H\), we have

$$\begin{aligned} \int _{S^{n-1}\cap H}\langle y, w\rangle ^2\textrm{d}\bar{\mu }(w)&=\int _{S^{n-1}\setminus H^{\perp }}\Big \langle y,\frac{\text {P}_Hu}{\Vert \text {P}_Hu\Vert }\Big \rangle ^2\Vert \text {P}_Hu\Vert ^2\textrm{d}\mu (u) \nonumber \\&=\int _{S^{n-1}}\langle y, u\rangle ^2\textrm{d}\mu (u)=\Vert y\Vert ^2. \end{aligned}$$
(3.4)

Thus, \(\bar{\mu }\) is isotropic on \(S^{n-1}\cap H\), and

$$\begin{aligned} \bar{\mu }(S^{n-1}\cap H)=\int _{S^{n-1}\setminus H^{\perp }}\Vert \text {P}_Hu\Vert ^2\textrm{d}\mu (u)=k. \end{aligned}$$
(3.5)

To prove Theorem 1.2, we shall use the continuous version of the Brascamp–Lieb inequality due to Barthe [4].

Theorem 3.1

(Brascamp–Lieb inequality) Let \(\mu \) be an isotropic Borel measure on \(S^{n-1}\) and let \((f_{u})_{u\in S^{n-1}}\) be a family of functions \(f_{u}:\mathbb {R}\rightarrow [0,+\infty )\) satisfying the following two conditions:

\(\mathrm {(I)}\) There exist a positive continuous function \(F:S^{n-1}\times \mathbb {R}\rightarrow (0,+\infty )\) and two functions f, g on \(S^{n-1}\) with \(f <g\) pointwise and f (resp. g) being either real-valued continuous or constant with value in \(\{-\infty ,+\infty \}\) such that for all \((u,t)\in S^{n-1}\times \mathbb {R}\)

$$\begin{aligned} f_{u}(t)=\textbf{1}_{t\in [f(u),g(u)]}F(u,t). \end{aligned}$$

\(\mathrm {(II)}\) There exists an integrable and bounded function U such that for all \(u\in S^{n-1}\), one has \(0\le f_{u}\le U\).

Then,

$$\begin{aligned} \int _{\mathbb {R}^n}\exp \left( \int _{S^{n-1}}\log f_{u}(\langle x,u\rangle )\textrm{d}\mu (u)\right) \textrm{d}x\le \exp \left( \int _{S^{n-1}}\log \left( \int _{\mathbb {R}}f_{u}\right) \textrm{d}\mu (u)\right) . \end{aligned}$$

Theorem 3.2

Let \(\mu \) be an even isotropic measure on \(S^{n-1}\) and \(C=\textrm{conv}\{\textrm{supp}\,\mu \}.\) Then, for any \(H\in G_{n,k}\),

$$\begin{aligned} \widetilde{\mathcal {W}}(\lambda (C^\circ \cap H))\le \Big (1+2\lambda \sqrt{\frac{n}{k}}\Big )^k=\mathcal {W}\Big (\lambda \sqrt{\frac{n}{k}}B_{\infty }^k\Big ), \ \ \ \lambda \ge 0. \end{aligned}$$

Proof

From (1.14), we have

$$\begin{aligned} \lambda (C^{\circ }\cap H)&=\Big \{x\in H: \langle x, u\rangle \le \lambda \ \ \text {for all}\ \ u\in \text {supp}\,\mu \Big \}\nonumber \\&=\Big \{x\in H: \langle x, \text {P}_Hu\rangle \le \lambda \ \ \text {for all}\ \ u\in \text {supp}\,\mu \setminus H^{\perp }\Big \}\nonumber \\&=\bigg \{x\in H: \Big \langle x,\frac{\textrm{P}_Hu}{\Vert \textrm{P}_Hu\Vert }\Big \rangle \le \frac{\lambda }{\Vert \textrm{P}_Hu\Vert } \ \ \ \text {for all} \ u\in \text {supp}\,\mu \setminus H^{\perp }\bigg \}. \end{aligned}$$

For \(w\in S^{n-1}\cap H\), denote by \(\langle w\rangle \) the 1-dimensional subspace spanned by w. We define the function \(F_{\lambda ,w}:\mathbb {R}\rightarrow [0,\infty )\) by

$$\begin{aligned} F_{\lambda ,w}(s)=e^{-\pi \widetilde{{d}}\big (sw, \textrm{P}_{\langle w\rangle }(\lambda (C^{\circ }\cap H))\big )^2}, \ \ \ s\in \mathbb {R}. \end{aligned}$$

It is easy to see that the function \(F_{\lambda ,w}(s)\) is positive continuous with respect to w on \(S^{n-1}\cap H\) and s on \((-\infty , +\infty )\), respectively. Moreover, \(0\le F_{\lambda ,w}(\cdot )\le 1\) for \(s\in \mathbb {R}\). So the function \(F_{\lambda ,w}(s)\) satisfies the conditions (I) and (II).

Notice that

$$\begin{aligned} \int _{\mathbb {R}}F_{\lambda ,w}(s)\textrm{d}s=\int _{\mathbb {R}}e^{-\pi \widetilde{d}\big (sw, \textrm{P}_{\langle w\rangle }(\lambda (C^{\circ }\cap H))\big )^2}\textrm{d}s=\widetilde{\mathcal {W}}(\textrm{P}_{\langle w\rangle }(\lambda (C^{\circ }\cap H))). \end{aligned}$$

If \(w=\frac{\text {P}_Hu}{\Vert \text {P}_Hu\Vert }\) for \(u\in \textrm{supp}\,\mu {\setminus } H^{\perp }\), then

$$\begin{aligned} \textrm{P}_{\langle w\rangle }(\lambda (C^{\circ }\cap H))\subseteq \Big [-\frac{\lambda }{\Vert \textrm{P}_Hu\Vert }\frac{\text {P}_Hu}{\Vert \text {P}_Hu\Vert },\frac{\lambda }{\Vert \textrm{P}_Hu\Vert }\frac{\text {P}_Hu}{\Vert \text {P}_Hu\Vert }\Big ]. \end{aligned}$$

It is easy to check that \(\Big [-\frac{\lambda }{\Vert \textrm{P}_Hu\Vert }\frac{\text {P}_Hu}{\Vert \text {P}_Hu\Vert },\frac{\lambda }{\Vert \textrm{P}_Hu\Vert }\frac{\text {P}_Hu}{\Vert \text {P}_Hu\Vert }\Big ]\) is an origin symmetric segment in \(l_{\frac{\text {P}_Hu}{\Vert \text {P}_Hu\Vert }}\). Thus, by the fact that \(\widetilde{\mathcal {W}}([-a,a])=1+2a\), we have

$$\begin{aligned} \widetilde{\mathcal {W}}(\textrm{P}_{\langle w\rangle }(\lambda (C^{\circ }\cap H)))\le \widetilde{\mathcal {W}}\Big (\Big [-\frac{\lambda }{\Vert \textrm{P}_Hu\Vert }\frac{\text {P}_Hu}{\Vert \text {P}_Hu\Vert },\frac{\lambda }{\Vert \textrm{P}_Hu\Vert }\frac{\text {P}_Hu}{\Vert \text {P}_Hu\Vert }\Big ]\Big )=1+\frac{2\lambda }{\Vert \textrm{P}_Hu\Vert }. \end{aligned}$$
(3.6)

Let \(\bar{\mu }\) be the measure on \(S^{n-1}\cap H\) defined in (3.3). Since \(\bar{\mu }\) is isotropic on \(S^{n-1}\cap H\), by Theorem 3.1, (3.3), (3.6), (3.5), the arithmetic–geometric mean inequality, Hölder’s inequality, (3.2), and again (3.5), we have

$$\begin{aligned}&\int _H\exp \bigg \{-\pi \int _{S^{n-1}\cap H}\widetilde{d}\big (\langle x,w\rangle w, \textrm{P}_{\langle w\rangle }(\lambda (C^{\circ }\cap H))\big )^2\textrm{d}\bar{\mu }(w)\bigg \}\textrm{d}x\nonumber \\&\le \exp \bigg \{\int _{S^{n-1}\cap H}\log (\int _{\mathbb {R}}F_{\lambda ,w}(s)ds)\textrm{d}\bar{\mu }(w)\bigg \} \nonumber \\ {}&=\exp \bigg \{\int _{S^{n-1}\cap H}\log \widetilde{\mathcal {W}}(\textrm{P}_{\langle w\rangle }(\lambda (C^{\circ }\cap H)))\textrm{d}\bar{\mu }(w)\bigg \}\nonumber \\&=\exp \bigg \{\int _{S^{n-1}\setminus H^{\perp }}\log \widetilde{\mathcal {W}}\Big (\textrm{P}_{\big \langle \frac{\text {P}_Hu}{\Vert \text {P}_Hu\Vert }\big \rangle }(\lambda (C^{\circ }\cap H))\Big )\Vert \text {P}_Hu\Vert ^2\textrm{d}\mu (u)\bigg \}\nonumber \\&\le \exp \bigg \{\int _{S^{n-1}\setminus H^{\perp }}\log \Big (1+\frac{2\lambda }{\Vert \textrm{P}_Hu\Vert }\Big )\Vert \text {P}_Hu\Vert ^2\textrm{d}\mu (u)\bigg \}\nonumber \\&\le \left( \frac{1}{k}\int _{S^{n-1}\setminus H^{\perp }} \Big (1+\frac{2\lambda }{\Vert \textrm{P}_Hu\Vert }\Big )\Vert \text {P}_Hu\Vert ^2\textrm{d}\mu (u)\right) ^k\nonumber \\&=\Big (1+\frac{2\lambda }{k}\int _{S^{n-1}\setminus H^{\perp }} \Vert \text {P}_Hu\Vert \textrm{d}\mu (u)\Big )^k\nonumber \\ {}&\le \left( 1+\frac{2\lambda }{k}\Big (\int _{S^{n-1}\setminus H^{\perp }}\textrm{d}\mu (u)\Big )^{\frac{1}{2}}\Big (\int _{S^{n-1}\setminus H^{\perp }}\Vert \textrm{P}_{H}u\Vert ^2\textrm{d}\mu (u)\Big )^{\frac{1}{2}}\right) ^k\nonumber \\&\le \left( 1+\frac{2\lambda }{k}\Big (\int _{S^{n-1}}\textrm{d}\mu (u)\Big )^{\frac{1}{2}} \Big (\int _{S^{n-1}\setminus H^{\perp }}\Vert \textrm{P}_{H}u\Vert ^2\textrm{d}\mu (u)\Big )^{\frac{1}{2}}\right) ^k \nonumber \\ {}&=\Big (1+2\lambda \sqrt{\frac{n}{k}}\Big )^k. \end{aligned}$$
(3.7)

On the other hand, for \(x\in H\), \(x_0\in \lambda (C^{\circ }\cap H)\cap \langle x \rangle \) and \(w\in \textrm{supp}\,\bar{\mu }\), we have

$$\begin{aligned} \widetilde{d}\big (\langle x,w\rangle w, \textrm{P}_{\langle w\rangle }(\lambda (C^{\circ }\cap H))\big )^2&\le \widetilde{d}\big (\langle x,w\rangle w, \langle x_0,w\rangle w\big )^2\\&=\langle x-x_0,w\rangle ^2, \end{aligned}$$

and by (3.4),

$$\begin{aligned} \int _{S^{n-1}\cap H}\widetilde{d}\big (\langle x,w\rangle w, \textrm{P}_{\langle w\rangle }(\lambda (C^{\circ }\cap H))\big )^2\textrm{d}\bar{\mu }(w)&\le \int _{S^{n-1}\cap H}\langle x-x_0,w\rangle ^2\textrm{d}\bar{\mu }(w)\\&=\Vert x-x_0\Vert ^2. \end{aligned}$$

Hence, for each \(x\in H\),

$$\begin{aligned} \int _{S^{n-1}\cap H}\widetilde{d}\big (\langle x,w\rangle w, \textrm{P}_{\langle w\rangle }(\lambda (C^{\circ }\cap H))\big )^2\textrm{d}\bar{\mu }(w)\le \widetilde{d}(x,\lambda (C^{\circ }\cap H))^2. \end{aligned}$$

Therefore, by (3.7) and the fact that \(\mathcal {W}(aB_{\infty }^n)=(1+2a)^n\) with \(a>0\), we finally have

$$\begin{aligned} \widetilde{\mathcal {W}}(\lambda (C^\circ \cap H))&=\int _He^{-\pi \widetilde{d}(x,\lambda (C^{\circ }\cap H))^2}\textrm{d}x\\ {}&\le \int _H\exp \bigg \{-\pi \int _{S^{n-1}\cap H}\widetilde{d}\big (\langle x,w\rangle w, \textrm{P}_{\langle w\rangle }(\lambda (C^{\circ }\cap H))\big )^2\textrm{d}\bar{\mu }(w)\bigg \}\textrm{d}x\\&\le \Big (1+2\lambda \sqrt{\frac{n}{k}}\Big )^k\\&=\mathcal {W}\Big (\lambda \sqrt{\frac{n}{k}}B_{\infty }^k\Big ), \end{aligned}$$

as desired. \(\square \)

Remark 3.3

Using the same approach in Theorem 3.2, we can also obtain the following inequality:

$$\begin{aligned} \mathcal {W}(\lambda (C^\circ \cap H))\le \mathcal {W}\Big (\lambda \sqrt{\frac{n}{k}}B_{\infty }^k\Big ). \end{aligned}$$

Thus, by the fact that \(\widetilde{\mathcal {W}}(\lambda (C^\circ \cap H))\le \mathcal {W}(\lambda (C^\circ \cap H))\), we are able to deduce Theorem 3.2 as well.