2.1 Background from differential geometry

For more information and the details in this section, we refer to, e.g., [Reference Gardner13, Reference Schneider29].

Let K be a convex body of class $C^2.$ For a point x on the boundary $\partial K$ of $K,$ we denote by $N(x)$ the unique outward unit normal vector of K at $x.$ The map $N_K: \partial K \to S^{n - 1}$ is called the spherical image map or Gauss map of K and is of class $C^1.$ Its differential is called the Weingarten map. The eigenvalues of the Weingarten map are the principal curvatures $k_i(x)$ of K at $x.$

The jth normalized elementary symmetric functions of the principal curvatures are denoted by $H_j$ . They are defined as follows:

(2.1) $$ \begin{align} H_j = {n - 1 \choose j}^{-1} \sum_{1\leq i_1 < \dots < i_j \leq n - 1} k_{i_1} \cdots k_{i_j,} \end{align} $$

for $j = 1, \dots , n-1$ and $H_0 = 1.$ Note that

$$ \begin{align*}H_1 = \frac{1}{n-1} \sum_{1\leq i \leq n - 1} k_{i}\end{align*} $$

is the mean curvature, that is, the average of principal curvatures, and

$$ \begin{align*}H_{n - 1}= \prod_{i=1}^{n-1} k_i\end{align*} $$

is the Gauss curvature.

We say that K is of class $C^2_+$ if K is of class $C^2$ and the Gauss map $\nu $ is a diffeomorphism. This means in particular that $N_K$ has a smooth inverse. This assumption is stronger than just $C^2,$ and is equivalent to the assumption that all principal curvatures are strictly positive, or that the Gauss curvature $H_{n-1} \ne 0.$ It also means that the differential of $N_K$ , i.e., the Weingarten map, is of maximal rank everywhere.

Let K be of class $C^2_+$ . For $u \in {\mathbb {R}}^n \setminus \{0\}$ , let $\xi _K(u)$ be the unique point on the boundary of K at which u is an outward normal vector. The map $\xi _K$ is defined on ${\mathbb {R}}^n \setminus \{0\}$ . Its restriction to the sphere $S^{n - 1},$ the map $\bar {\xi }_K: S^{n - 1} \to \partial K$ , is called the reverse spherical image map, or reverse Gauss map. The differential of $\bar {\xi }_K$ is called the reverse Weingarten map. The eigenvalues of the reverse Weingarten map are called the principal radii of curvature $r_1, \dots , r_{n - 1}$ of K at $u\in S^{n - 1}.$

The jth normalized elementary symmetric functions of the principal radii of curvature are denoted by $s_j$ . In particular, $s_0=1$ , and for $1 \leq j \leq n-1,$ they are defined by

(2.2) $$ \begin{align} s_j = {n - 1 \choose j}^{-1} \sum_{1\leq i_1 < \dots < i_j \leq n - 1} r_{i_1} \cdots r_{i_j}. \end{align} $$

Note that the principal curvatures are functions on the boundary of K and the principal radii of curvature are functions on the sphere.

Now, we describe the connection between $H_j$ and $s_j.$ For a body K of class $C^2_+$ , we have for $u\in S^{n - 1}$ that $\bar {\xi }_K(u) = N_K^{-1}(u)$ . In particular, the principal radii of curvature are reciprocals of the principal curvatures, that is,

$$ \begin{align*}r_i(u) = \frac1{k_i(\bar{\xi}_K(u))}. \end{align*} $$

This implies that for $x \in \partial K$ with $N_K(x)=u$ ,

$$ \begin{align*}s_j = {n - 1 \choose j}^{-1} \sum_{1\leq i_1 < \dots < i_j \leq n - 1} \frac1{k_{i_1}(\bar{\xi}_K(u)) \cdots k_{i_j}(\bar{\xi}_K(u))} = \frac{H_{n - 1 -j}}{H_{n - 1}}\Big(\bar{\xi}_K(u)\Big), \end{align*} $$

and

$$ \begin{align*}H_j = \frac{s_{n - 1 -j}}{s_{n - 1}}\Big(N_K(x)\Big), \end{align*} $$

for $j = 1, \dots , n-1.$

2.2 Definitions

For $\alpha \in \mathbb {R}$ and $k \in \mathbb {N}$ , the generalized binomial coefficients are defined as

(2.3) $$ \begin{align} {\alpha \choose k} = \left\{\vphantom{\begin{array}{@{\!\!\!\!\!\!\!\!}lll}&1, &\qquad\text{if }\, k=0,\\[2pt] &0, &\qquad\text{if }\, k<0 \, \text{ or }\, \alpha =0,\\[4pt] &\dfrac{\alpha(\alpha - 1)\cdots(\alpha - k + 1)}{k!}, &\qquad\text{if }\, k> 0. \end{array}_{}}\begin{array}{@{\kern-1pt\!\!\!\!\!\!}lll}&1, &\quad\text{if }\, k=0,\\[4pt] &0, &\quad\text{if }\, k<0 \, \text{ or }\, \alpha =0,\\[4pt] &\dfrac{\alpha(\alpha - 1)\cdots(\alpha - k + 1)}{k!}, &\quad\text{if }\, k> 0. \end{array}\right. \end{align} $$

For fixed $k \in \mathbb {N}$ , $m\in \mathbb {N} \cup \{0\}$ and fixed sequence $\vec {i}=\{i_j\}_{j=0}^{n-1}$ such that $i_1+2 i_2+ \cdots (n-1) i_{n-1} =m$ and all $p \in \mathbb {R}$ , $p \neq -n$ , we define

(2.4) $$ \begin{align} \omega_{m,k,\vec{i}}^p (K) = \int\limits_{\partial K}\frac{H_{n-1}(x)^{\frac{p}{n+p}}} {\langle x,N(x)\rangle ^{\frac{n(p-1)}{n+p}}} \,\langle x,N(x)\rangle ^{m-k} \, \prod\limits_{j = 1}^{n - 1}\left\{ {n - 1\choose j}^{i_j} H_{j}^{i_j}(x)\right\} \, d\mathcal{H}^{n-1}(x) \end{align} $$

and

(2.5) $$ \begin{align} u_{m,k,\vec{i}}^p (K) = \int\limits_{S^{n-1}}\frac{f_{K}(u)^{\frac{n}{n+p}}} {h_K(u) ^{\frac{n(p-1)}{n+p}}} \, h_K(u) ^{m-k} \, \frac{ \prod\limits_{j = 1}^{n - 1} {n - 1\choose j}^{i_j} s_{n-1-j}^{i_j} (u)}{s_{n-1}^{\sum\limits_j i_j+\frac p{n+p}}(u)} \, d\mathcal{H}^{n-1}(u). \end{align} $$

Remark 2.1 Note that the above quantities are vanishing for polytopes. Therefore, we will treat only $C^2_+$ convex bodies throughout the text and then

(2.6) $$ \begin{align} \omega_{m,k,\vec{i}}^p (K)= u_{m,k,\vec{i}}^p (K). \end{align} $$

Denote

$$ \begin{align*}c(n) = \prod\limits_{j = 1}^{n - 1} {n - 1\choose j}^{i_j}. \end{align*} $$

Let $\mu _{m,k,\vec {i}}$ be the measure on $\partial K$ with density

$$ \begin{align*}d\mu_{m,k,\vec{i}} (x) = c(n) \, \langle x,N(x)\rangle^{m-k} \prod\limits_{j = 1}^{n - 1} H_{j}^{i_j}(x)\, d\mathcal{H}^{n-1}(x), \end{align*} $$

with respect to the surface measure $\mathcal {H}^{n-1}$ on $\partial K,$ and let $\nu _{m,k,\vec {i}}$ be the measure on $S^{n-1}$ with density

$$ \begin{align*}d\nu_{m,k,\vec{i}} (u) =c(n) \, h^{m-k}_K(u) \frac{\prod\limits_{j = 1}^{n - 1}s_{n-1-j}^{i_j} (u)}{s_{n-1}^{\sum\limits_j i_j+\frac p{n+p}}(u)} d\mathcal{H}^{n-1}(u) \end{align*} $$

with respect to the surface measure $\mathcal {H}^{n-1}$ on $S^{n-1}$ . To keep notations simple, we mostly write $\mu _{\vec {i}}$ and $\nu _{\vec {i}}$ instead of $\mu _{m,k,\vec {i}}$ and $\nu _{m,k,\vec {i}}$ .

We then define the weighted $L_p$ -affine surface areas by

(2.7) $$ \begin{align} \mu_{\vec{i}} - as_p(K)= \omega_{m,k,\vec{i}}^p (K) = \int\limits_{\partial K} \frac{H_{n-1}(x)^{\frac{p}{n+p}}} {\langle x,N(x)\rangle ^{\frac{n(p-1)}{n+p}}} d\mu_{\vec{i}}(x). \end{align} $$

Definition (1.2) explains that those can be considered as $L_p$ -affine surface area weighted by the measure $\mu _{\vec {i}}$ . In particular,

$$ \begin{align*}\omega_{m,k,\vec{i}}^0 (K) = \int\limits_{\partial K} \langle x,N(x)\rangle d\mu_{\vec{i}}(x) = n(\mu_{\vec{i}}-{\mathrm{vol}}_n(K)) \end{align*} $$

is a weighted volume of K, weighted by the measure $\mu _{\vec {i}}$ and

$$ \begin{align*}\omega_{m,k,\vec{i}}^\infty (K) = \int\limits_{\partial K} \frac{H_{n-1}}{\langle x,N(x)\rangle^n} d\mu_{\vec{i}}(x) =n( \mu_{\vec{i}} - {\mathrm{vol}}_n(K^\circ)) \end{align*} $$

is a weighted volume of $K^\circ $ , weighted by the measure $\mu _{\vec {i}}$ , where

$$ \begin{align*}K^\circ=\{ y \in \mathbb{R}^n: \langle y, x \rangle \leq 1, \text{ for all } x \in K \} \end{align*} $$

is the polar body of K.

We can take another point of view for the measure $d\mathcal {H}^{n-1}$ on $S^{n-1}$ via the density $f_p(K,u) = \left (\frac {f_K(u)}{h_K^{p-1}(u)}\right )^{\frac n{n+p}}$ . This density was introduced by Lutwak [Reference Lutwak23]. Then

$$ \begin{align*}u_{m,k,\vec{i}}^p (K) = c(n) \int\limits_{S^{n-1}} f_p(K,u) h_K(u) ^{m-k} \, \frac{ \prod\limits_{j = 1}^{n - 1} s_{n-1-j}^{i_j} (u)}{s_{n-1}^{\sum\limits_j i_j+\frac p{n+p}}(u)} \, d\mathcal{H}^{n-1}(u). \end{align*} $$

2.3 Special cases

Note that

$$ \begin{align*}\omega_{0, 0, 0}^p(K) = u_{0,0,0}^p(K) = as_p(K), \end{align*} $$

where we denote the sequence $\vec {i} = \{0, \dots , 0\}$ as $0$ .

1. When $k = m$ , we get

$$ \begin{align*}\omega_{m,m,\vec{i}}^p (K) = \int\limits_{\partial K}\frac{H_{n-1}(x)^{\frac{p}{n+p}}} {\langle x,N(x)\rangle ^{\frac{n(p-1)}{n+p}}} \, \prod\limits_{j = 1}^{n - 1}\left\{ {n - 1\choose j}^{i_j} H_{j}^{i_j}(x)\right\} \, d\mathcal{H}^{n-1}(x). \end{align*} $$

2. $m = 0$ implies that $\vec {i} = 0$ and (2.4) simplifies to

$$ \begin{align*}\omega_{0,k,0}^p (K) = \int\limits_{\partial K}\frac{H_{n-1}(x)^{\frac{p}{n+p}}} {\langle x,N(x)\rangle ^{\frac{n(p-1)}{n+p}+k}} \, d\mathcal{H}^{n-1}(x) = as_{p + \frac kn (n+p),\, -k}(K) \end{align*} $$

(see [Reference Tatarko and Werner34, equation (29)]).

3. For the Euclidean unit ball $B^n_2$ , we get

(2.8) $$ \begin{align} \omega_{m,k,\vec{i}}^p (B^n_2) = {\mathrm{vol}}_{n-1}(B^n_2) \prod\limits_{j = 1}^{n - 1} {n - 1\choose j}^{i_j} = c(n)\, {\mathrm{vol}}_{n-1}(\partial B^n_2), \end{align} $$

which does not depend on k. Note that if $i_0 \ne 0$ and $i_1 = \dots = i_{n-1} = 0$ , that is, ${\vec {i} = \{i_0, 0, \dots , 0\}}$ , then $\omega _{m,k,\vec {i}}^p (B^n_2) = {\mathrm {vol}}_{n-1}(\partial B^n_2)$ .

4. If $p = 0$ , we have

$$ \begin{align*}\omega_{m,k, \vec{i}}^0(K) = c(n) \int\limits_{\partial K} \langle x,N(x)\rangle ^{m-k+1} \, \prod\limits_{j = 1}^{n - 1}H_{j}^{i_j}(x) \, d\mathcal{H}^{n-1}(x). \end{align*} $$

When in addition $k = m$ , we get

$$ \begin{align*}\omega_{m,m, \vec{i}}^0(K) = c(n) \int\limits_{\partial K} \langle x,N(x)\rangle \, \prod\limits_{j = 1}^{n - 1} H_{j}^{i_j}(x) \, d\mathcal{H}^{n-1}(x). \end{align*} $$

5. If $p = \infty $ , we get that

$$ \begin{align*}\omega_{m,m, \vec{i}}^\infty(K) = c(n) \int\limits_{\partial K} \frac{H_{n-1}}{\langle x,N(x)\rangle^n} \, \prod\limits_{j = 1}^{n - 1} H_{j}^{i_j}(x) \, d\mathcal{H}^{n-1}(x). \end{align*} $$

3.1 Valuation, invariance, and homogeneity

Proof The proof follows immediately from results in [Reference Tatarko and Werner35]. We present an outline of the proof for completeness.

1. Valuation. As was shown in [Reference Tatarko and Werner35, Theorem 5.9], for all $1 \leq i_1, \dots , i_{n-1} \leq n-1$ , $1\leq j \leq n-1$ , and $\alpha _1, \dots , \alpha _{j} \geq 0$ ,

$$ \begin{align*}\int\limits_{\partial K}\frac{H_{n-1}(x)^{\frac{p}{n+p}}} {\langle x,N(x)\rangle ^{\frac{n(p-1)}{n+p}}} \,\langle x,N(x)\rangle ^{k-m} \, \prod\limits_{i = 1}^{j} k_{i_j}^{\alpha_j}\, d\mathcal{H}^{n-1}(x) \end{align*} $$

is a valuation. It immediately follows that $ \omega _{m,k,\vec {i}}^p (K) $ are valuations as the linear combination of valuations is again a valuation.

2. Homogeneity. Similarly to [Reference Tatarko and Werner35, Theorem 5.1] we can show that $\omega _{m, k, \vec {i}}^p (K)$ are homogeneous of order $n\frac {n-p}{n+p} - k$ . Applying [Reference Tatarko and Werner35, Proposition 5.4] with $T = a\, Id$ , we get that $\omega _{m, k, \vec {i}}^p (K) = a^{k - n\frac {n-p}{n+p}} \omega _{m, k, \vec {i}}^p (aK)$ .

3. Invariance. If T is a rotation or a reflection, then $|\det T|=1$ , $\|T^{-1t}(N_{ K}(T^{-1}(y)))\| = \| N_{ K}(T^{-1}(y))\| = 1$ and for all $1 \leq j \leq n-1$ ,

$$ \begin{align*}\{ H_{j}(y): y \in \partial T(K) \}= \{H_{j}\left(x \right): \, x \in \partial K \}.\end{align*} $$

Thus, using these observations and [Reference Tatarko and Werner35, Proposition 5.4], we get

$$ \begin{align*} &\omega_{m, k, \vec{i}}^p (K) =\\ & \int\limits_{\partial T(K)} \langle y, N_{T(K)}(y) \rangle^{m - k + \frac{n(1-p)}{n+p}} H_{n-1}^{\frac p{n+p}} (y) \prod\limits_{j = 1}^{n - 1} {n - 1\choose j}^{i_j} H_{j}^{i_j}\left(y \right) \, d\mathcal{H}^{n-1}(y) = \omega_{m, k, \vec{i}}^p (T(K)). \end{align*} $$

3.2 Inequalities

Proof (i) By Hölder’s inequality—which enforces the condition $\frac {(n+r)(s-t)}{(n+t)(s-r)}>1$ , we then get

(3.1) $$ \begin{align} \mu_{\vec{i}} - as_r(K) &= \omega_{m,k,\vec{i}}^r (K) \\ &= \int\limits_{\partial K}\frac{H_{n-1}(x)^{\frac{r}{n+r}}} {\langle x,N(x)\rangle ^{\frac{n(r-1)}{n+r}}} \,\langle x,N(x)\rangle ^{m-k} \, \prod\limits_{j = 1}^{n - 1}\left\{ {n - 1\choose j}^{i_j} H_{j}^{i_j}(x)\right\} \, d\mathcal{H}^{n-1}(x) \nonumber \\ &= \int _{\partial K} \left(\frac{H_{n-1}(x)^{\frac{t}{n+t}}}{\langle x, N(x)\rangle ^{\frac{n(t-1)}{n+t}}} \right)^{\frac{(r-s)(n+t)}{(t-s)(n+r)}} \left(\frac{H_{n-1}(x)^{\frac{s}{n+s}}}{\langle x, N(x)\rangle ^{\frac{n(s-1)}{n+s}}}\right)^{\frac{(t-r)(n+s)}{(t-s)(n+r)}}\, d\mu_{\vec{i}}(x) \nonumber \\ &\leq \big(\omega_{m,k,\vec{i}}^t (K) \big)^{\frac{(r-s)(n+t)}{(t-s)(n+r)}} \big(\omega_{m,k,\vec{i}}^s (K)\big)^{\frac{(t-r)(n+s)}{(t-s)(n+r)}} \nonumber \\ &= \big(\mu_{\vec{i}} - as_t(K) \big)^{\frac{(r-s)(n+t)}{(t-s)(n+r)}} \big(\mu_{\vec{i}} - as_s(K) \big)^{\frac{(t-r)(n+s)}{(t-s)(n+r)}}. \nonumber \end{align} $$

(ii) Similarly, again using Hölder’s inequality—which now enforces the condition $~\frac {(n+r)t}{(n+t)r}>1,~$

$$ \begin{align*} \mu_{\vec{i}} - as_r(K) &=\omega_{m,k,\vec{i}}^r(K)\\ &=\int\limits_{\partial K} \frac{H_{n-1}(x)^{\frac{r}{n+r}}}{\langle x, N(x)\rangle ^{\frac{n(r-1)}{n+r}}}\, d\mu_{\vec{i}} (x) = \int\limits_{\partial K} \left(\frac{H_{n-1}(x)^{\frac{t}{n+t}}}{\langle x, N(x)\rangle ^{\frac{n(t-1)}{n+t}}} \right)^{\frac{r(n+t)}{t(n+r)}} \frac{d\mu_{\vec{i}} (x)}{\langle x, N(x)\rangle ^{\frac{(r-t)n}{(n+r)t}}} \\ &\leq \big(\omega_{m,k,\vec{i}}^t(K)\big)^{\frac{r(n+t)}{t(n+r)}} \big( \omega_{m,k,\vec{i}}^0(K)\big)^{\frac{(t-r)n}{(n+r)t}} \nonumber \\ &= \bigg(\mu_{\vec{i}} - as_t(K)\bigg)^{\frac{r(n+t)}{{t(n+r)}}} \bigg(n(\mu_{\vec{i}}-{\mathrm{vol}}_n(K))\bigg)^{\frac{n(t-r)}{{t(n+r)}}}. \end{align*} $$

The equality characterizations follow from the equality characterization of Hölder’s inequality.

Equality holds in (i) and (ii) if and only if equality holds in Hölder’s inequality which happens if and only if

$$ \begin{align*}\frac{H_{n-1}}{\langle x,N(x)\rangle^{n+1}} = \mathrm{constant} \end{align*} $$

$\mu _{\vec {i}}$ -almost everywhere on $\partial K$ . As $\partial K$ is $C^2_+$ , $\{ x \in \partial K: \mu _{\vec {i}}(x) =0\} = \emptyset $ and therefore

$$ \begin{align*}\frac{H_{n-1}}{\langle x,N(x)\rangle^{n+1}} = \mathrm{constant} \end{align*} $$

holds for all $x \in \partial K$ . Thus, we can use the following theorem by Petty [Reference Petty27], which then finishes the proof.

Theorem 3.3 [Reference Petty27]

Let K be a convex body in ${\mathbb {R}}^n$ that is $C^2_+$ . K is an ellipsoid if and only if, for all x in $\partial K,$

$$ \begin{align*}\frac{H_{n-1}}{\langle x,N(x)\rangle^{n+1}} = c, \end{align*} $$

where $c>0$ is a constant.

Theorem 3.4 Let K be a $C^2_+$ convex body in $\mathbb R^n$ with centroid at the origin. If $r<s<k,$ then

$$\begin{align*}\omega_{m,s,\vec{i}}^p (K) \leq (\omega_{m,k,\vec{i}}^p (K))^{\frac{k-s}{k-r}} (\omega_{m,r,\vec{i}}^p (K))^{\frac{s-r}{k-r.}} \end{align*}$$

Equality holds if and only if K is a ball.

Proof

(3.2) $$ \begin{align} &\omega_{m,s,\vec{i}}^p (K) =\int\limits_{\partial K}\frac{H_{n-1}(x)^{\frac{p}{n+p}}} {\langle x,N(x)\rangle ^{\frac{n(p-1)}{n+p}}} \,\langle x,N(x)\rangle ^{m-s} \, \prod\limits_{j = 1}^{n - 1}\left\{ {n - 1\choose j}^{i_j} H_{j}^{i_j}(x)\right\} \, d\mathcal{H}^{n-1}(x) \nonumber \\ &= \int\limits_{\partial K} \left(\frac{H_{n-1}(x)^{\frac{p}{n+p}}}{\langle x, N(x)\rangle ^{\frac{n(p-1)}{n+p}}} \langle x,N(x)\rangle ^{m-k} c(n) \prod\limits_{j = 1}^{n - 1} H_{j}^{i_j}\right)^{\frac{s-r}{k-r}}\\ &\hskip 10mm \cdot\left(\frac{H_{n-1}(x)^{\frac{p}{n+p}}}{\langle x, N(x)\rangle ^{\frac{n(p-1)}{n+p}}} \langle x,N(x)\rangle ^{m-r} c(n) \prod\limits_{j = 1}^{n - 1} H_{j}^{i_j}\right)^{\frac{k-s}{k-r}} d\mathcal{H}^{n-1}(x) \nonumber\\ &\leq \big(\omega_{m,k,\vec{i}}^p (K) \big)^{\frac{s-r}{k-r}} \big(\omega_{m,r,\vec{i}}^p (K) \big)^{\frac{k-s}{k-r}}. \nonumber \end{align} $$

Equality holds if and only if equality holds in Hölder’s inequality which happens if and only if $\langle x, N(x)\rangle = \mathrm {constant}$ if and only if K is a ball.

We also obtain monotonicity behaviors.

3.3 Geometric interpretations

We recall several constructions of convex bodies associated with a given convex body K. Namely:

1. Weighted floating bodies [Reference Werner36]

Let K be a convex body in $ {\mathbb R}^n$ and denote by $\lambda $ the Lebesgue measure on $\mathbb {R}^n$ . Let $s \geq 0,$ and let $f:K\rightarrow \mathbb R$ be an integrable function such that $f>0\ \lambda $ -a.e. The weighted floating body $F(K,f,s)$ was defined in [Reference Werner36] (see also [Reference Besau, Ludwig and Werner3–Reference Besau and Werner4]) as the intersection of all closed half-spaces $H^{+}$ whose defining hyperplanes H cut off a set of $(f\, \lambda )$ -measure less than or equal to s from K,

$$ \begin{align*} F(K,f,s)=\bigcap_{\int_{ K\cap H^{-}} f\, d\lambda \leq s}H^{+}. \end{align*} $$

It was shown in [Reference Werner36] that

$$ \begin{align*}2 \bigg(\frac{\mbox{vol}_{n-1}(B^{n-1}_2)}{n+1}\bigg)^{\frac{2}{n+1}} \lim_{s \to 0^+} \frac{\mbox{vol}_n(K)-\mbox{vol}_n(F(K,f,s))} {s^{\frac{2}{n+1}}}= \int_{\partial K} \frac{H_{n-1}^{\frac{1}{n+1}}} {f^{\frac{2}{n+1}}} \, d\mathcal{H}^{n-1}. \end{align*} $$

2. Surface bodies [Reference Schütt and Werner33]

Let K be a convex body, and let $f:\partial K\rightarrow \mathbb R$ be a nonnegative, integrable function with $\int _{\partial K}f \, \mathcal {H}^{n-1}=1$ . The probability measure $\mathbb P_{f}$ is the measure on $\partial K$ with density f. Let $s \geq 0$ . The surface body $S(K, f,s)$ was defined in [Reference Schütt and Werner33] as the intersection of all the closed half-spaces $H^{+}$ whose defining hyperplanes H cut off a set of $\mathbb P_{f}$ -measure less than or equal to s from $\partial K$ ,

$$ \begin{align*} S(K,f,s)=\bigcap_{\mathbb P_{f}(\partial K\cap H^{-})\leq s}H^{+}. \end{align*} $$

It was shown in [Reference Schütt and Werner33] that

$$ \begin{align*}2 \bigg(\mbox{vol}_{n-1}(B^{n-1}_2)\bigg)^{\frac{2}{n-1}}\lim_{s \to 0^+} \frac{\mathrm{vol}_n(K)-\mathrm{vol}_n(S(K, f,s))} {s^{\frac{2}{n-1}}}= \int_{\partial K} \frac{H_{n-1}^{\frac{1}{n-1}}} {f^{\frac{2}{n-1}}} \, d\mathcal{H}^{n-1}. \end{align*} $$

3. Random polytopes [Reference Schütt and Werner32]

A random polytope is the convex hull of finitely many points that are chosen with respect to a probability measure. In [Reference Schütt and Werner32], random polytopes are considered where the points are chosen from $\partial K$ with respect to $\mathbb P_{f}$ , where $f:\partial K\rightarrow \mathbb R$ is an integrable, nonnegative function with $\int _{\partial K} f \, d\mathcal {H}^{n-1} =1$ and $ d \,\mathbb {P}_{f} = f \,d\mathcal {H}^{n-1}$ . Then the expected volume of such a random polytope is

$$ \begin{align*}\mathbb E(f,N)=\mathbb E(\mathbb P_{f},N)=\int_{\partial K}\cdots\int_{\partial K}\mathrm{vol}_{n}([x_{1},\dots,x_{N}]) d\mathbb P_f(x_{1})\dots d\mathbb P_f(x_{N}), \end{align*} $$

where $[x_{1},\dots ,x_{N}]$ is the convex hull of the points $x_{1},\dots ,x_{N}$ . It was shown in [Reference Schütt and Werner32] that under mild smoothness assumptions on K,

$$ \begin{align*}\lim_{N \to \infty} \frac{\mathrm{vol}_n(K)-\mathbb{E}(f ,N)}{\left(\frac{1}{N}\right)^{\frac{2}{n-1}}}= c_n \int_{\partial K} \frac{H_{n-1}^{\frac{1}{n-1}}} {f^{\frac{2}{n-1}}} \, d\mathcal{H}^{n-1}, \end{align*} $$

where $c_n=\frac {(n-1)^{\frac {n+1}{n-1}}\Gamma \left (n+1+\tfrac {2}{n-1}\right )} {2(n+1)!(\mathrm {vol}_{n-2}(\partial B_{2}^{n-1}))^{\frac {2}{n-1}}}.$

4. Ulam floating bodies [Reference Huang, Slomka and Werner19]

Given a Borel measure $\mu $ on ${\mathbb {R}}^{n}$ , the metronoid associated with $\mu $ was introduced by Huang and Slomka [Reference Huang and Slomka18] and is the convex set defined by

$$\begin{align*}\mathcal{M}(\mu)=\bigcup_{\substack{0\leq f\leq1,\\ \int_{{\mathbb{R}}^{n}}f d{\mu}=1}} \left\{ \int_{{\mathbb{R}}^{n}}yf\left(y\right) d{\mu}\left(y\right)\right\} \!, \end{align*}$$

where the union is taken over all functions $0\le f\le 1$ for which $\int _{{\mathbb {R}}^{n}}f d{\mu =1}$ and $\int _{{\mathbb {R}}^{n}}yf\left (y\right )d{\mu }\left (y\right )$ exists. The metronoid $\mathcal {M}_{\delta }(K)$ is generated by the uniform measure on K with total mass $\delta ^{-1}\left |K\right |$ . Namely, let be the characteristic function of K, and let $\mu $ be the measure whose density with respect to Lebesgue measure is . Then $\mathcal {M}_{\delta }(K):=\mathcal {M}(\mu )$ . In [Reference Huang, Slomka and Werner19] weighted variations of $\mathcal {M}_{\delta }(K)$ were defined where the weight is given by a positive continuous function $f:K\to {\mathbb {R}}$ ,

It was shown in [Reference Huang, Slomka and Werner19] that

$$ \begin{align*} \lim_{\delta\rightarrow 0^+}\frac{ \mathrm{vol}_{n}\left (K\right)- \mathrm{vol}_{n}\left(\mathcal{M}_{\delta}(K, f)\right)}{\delta^{\frac{2}{n+1}}}= 2\frac{n+1}{n+3}\left(\frac{{\mathrm{vol}}_{n-1}(B_{2}^{n-1})}{n+1}\right)^{\frac{2}{n+1}} \int_{\partial K} \frac{H_{n-1}^{\frac{1}{n+1}}} {f^{\frac{2}{n+1}}} \, d\mathcal{H}^{n-1}. \end{align*} $$

This leads to geometric interpretations of the weighted $L_p$ -affine surface areas in terms of these associated bodies. That is, if we let

$$ \begin{align*}f(x)= \frac{H_{n-1}(x)^{\frac{n(1-p)}{2(n+p)}}} {\left[c(n)\, H_{j}^{i_j}(x)\right]^{\frac{n+1}{2}}} \,\langle x,N(x)\rangle ^{\left(\frac{n(p-1)}{n+p} +k-m\right) \frac{n+1}{2} } \end{align*} $$

in the case of weighted floating bodies and Ulam floating bodies, and

$$ \begin{align*}f(x)= \frac{H_{n-1}(x)^{\frac{2p+n(1-p)}{2(n+p)}}} {\left[c(n)\, H_{j}^{i_j}(x)\right]^{\frac{n-1}{2}}} \,\langle x,N(x)\rangle ^{\left(\frac{n(p-1)}{n+p} +k-m\right) \frac{n-1}{2} } \end{align*} $$

in the case of surface bodies and random polytopes, respectively, and denote by $K_{f,s}$ the corresponding associated body, i.e.,

$$ \begin{align*}K_{f,s}= F(K, f,s), \ \ \text{or} \ \ K_{f,s}= S(K, f,s), \ \ \text{or} \ \ K_{f,s}= \mathcal{M}_{s}(K, f), \ \ \text{or} \ \ {\mathrm{vol}}_n(K_{f,s}) = \mathbb E(f,N), \end{align*} $$

then, with the properly adjusted constant $c_n$ , we get the following proposition.

Proposition 3.7 Let K be a $C^2_+$ convex body in ${\mathbb {R}}^n$ with centroid at the origin. Let $p \ne -n$ be a real number.

$$ \begin{align*} c_n \lim_{s \to 0} \frac{\mathrm{vol}_n(K)-\mathrm{vol}_n(K_{f,s})} {s^{\frac{2}{n-1}}}= \omega_{m,k,\vec{i}}^p (K). \end{align*} $$

3.4 Asymptotics for the weighted $L_p$ -affine surface areas

Let $(X, \mu )$ be a measure space, and let $dP = p d\mu $ and $Q = q d\mu $ be measures in X that are absolutely continuous with respect to the measure $\mu $ . The Kullback–Leibler divergence or relative entropy from P to Q is defined as in [Reference Cover and Thomas11]

(3.3) $$ \begin{align} D_{KL}(P||Q) = \int\limits_X p \log{\frac pq} d\mu. \end{align} $$

The information inequality (also called Gibb’s inequality) [Reference Cover and Thomas11] holds for the Kullback–Leibler divergence. Let P and Q be as above, then

(3.4) $$\begin{align} D_{KL}(P||Q) \geq 0 \end{align}$$

with equality if and only if $P=Q$ .

We will apply this when $(X, \mu ) = (\partial K, \mu _{\vec {i}})$ , where K is $C^2_+$ and densities p and q with respect to $\mu _{\vec {i}}$ given by

(3.5) $$ \begin{align} p_K(x) = \frac{H_{n-1}}{\langle x,N(x)\rangle^n}, \qquad q_K(x) = \langle x,N(x)\rangle. \end{align} $$

We let

(3.6) $$ \begin{align} P_K = \frac{H_{n-1}}{\langle x,N(x)\rangle^n} \mu_{\vec{i}}, \qquad Q_K = \langle x,N(x)\rangle \mu_{\vec{i}}. \end{align} $$

Recall that classical cone measure $cm_K$ on $\partial K$ is defined as follows: for every measurable set $A \subseteq \partial K,$

$$ \begin{align*}cm_K(A) = {\mathrm{vol}}_n(\{t a: \ a \in A,\, t\in [0, 1]\}). \end{align*} $$

It is wellknown (see, e.g., [Reference Paouris and Werner26] for a proof) that the measures $\frac {H_{n-1}}{\langle x,N(x)\rangle ^n}\mathcal {H}^{n-1}$ and $\langle x,N(x)\rangle \mathcal {H}^{n-1}$ are the cone measures of K and $K^\circ $ ,

$$ \begin{align*} cm_K(A) = \frac1n \int\limits_{A} \langle x,N(x)\rangle \mathcal{H}^{n-1}(x), \qquad cm_{K^\circ}(A) = \frac1n \int\limits_{A} \frac{H_{n-1}}{\langle x,N(x)\rangle^n} \mathcal{H}^{n-1}(x). \end{align*} $$

The interpretation of $cm_{K^\circ }$ as the “cone measure of $K^\circ $ ” is via the Gauss map on $K^\circ $ , $N_{K^\circ }: \partial K^\circ \rightarrow S^{n-1}, \ x \longmapsto N_{K^\circ }(x)$ and the inverse of the Gauss map on $K N_{K}: \partial K \rightarrow S^{n-1}, \ x \longmapsto N_{K}(x)$ ,

$$ \begin{align*}N_K^{-1}N_{K^\circ} cm_{K^\circ,} \end{align*} $$

where we use $N_K(x)$ to emphasize that it is the normal vector of a body K at $x \in \partial K$ .

We now define the weighted cone measures $\mu _{\vec {i}} -cm_K$ of K and $\mu _{\vec {i}} -cm_{K^\circ }$ of $K^\circ $ by

(3.7) $$ \begin{align} \mu_{\vec{i}} -cm_K = \frac1n \int\limits_{A} \langle x,N(x)\rangle \mu_{\vec{i}}(x), \end{align} $$

(3.8) $$ \begin{align} \mu_{\vec{i}} -cm_{K^\circ} (A )= \frac1n \int\limits_{A} \frac{H_{n-1}}{\langle x,N(x)\rangle^n} \mu_{\vec{i}}(x). \end{align} $$

We show that the limits of the weighted $L_p$ affine surface areas are entropy powers.

Theorem 3.8 Let K be a $C^2_+$ convex body in $\mathbb R^n$ with centroid at the origin.

(i)

$$ \begin{align*} \lim\limits_{p \rightarrow \infty} \left(\frac{\omega_{m, k, \vec{i}}^p (K)}{\omega^\infty_{m, k, \vec{i}}(K)}\right)^{n+p} = \mathrm{exp} \left(-\frac{n \, D_{KL} (P_K||Q_K)}{\mu_{\vec{i}} - {\mathrm{vol}}_n(K^\circ)}\right). \end{align*} $$

(ii)

$$ \begin{align*} \lim\limits_{p \rightarrow 0} \left(\frac{\omega_{m, k, \vec{i}}^p (K^\circ)}{\omega^0_{m, k, \vec{i}}(K^\circ)}\right)^{\frac{n(n+p)}p} = \mathrm{exp} \left(-\frac{n \, D_{KL} (P_{K^\circ}||Q_{K^\circ})}{\mu_{\vec{i}} - {\mathrm{vol}}_n(K^\circ)}\right). \end{align*} $$

Proof (i) We use L’Hospital’s rule

$$ \begin{align*} &\lim\limits_{p \rightarrow \infty} \ln \left(\left(\frac{\omega_{m, k, \vec{i}}^p (K)}{\omega^\infty_{m, k, \vec{i}}(K)}\right)^{n+p} \right) = \lim\limits_{p \rightarrow \infty} \frac{\ln \left(\frac{\omega_{m, k, \vec{i}}^p (K)}{\omega^\infty_{m, k, \vec{i}}(K)} \right)}{(n+p)^{-1}} = - \lim\limits_{p \rightarrow \infty} (n+p)^2 \frac{\frac{d}{dp} \left(\omega_{m, k, \vec{i}}^p (K)\right) }{\omega_{m, k, \vec{i}}^p (K)} \\ &= -\lim\limits_{p \rightarrow \infty} \frac{(n+p)^2}{\omega_{m, k, \vec{i}}^p (K)} \int\limits_{\partial K} \frac{d}{dp} \left( \mathrm{exp} \left( \ln \left( \frac{H_{n-1}(x)^{\frac{p}{n+p}}} {\langle x,N_K(x)\rangle ^{\frac{n(p-1)}{n+p}}} \right) \right) \right) d\mu_{\vec{i}} (x)\\ &= -\lim\limits_{p \rightarrow \infty} \frac{n}{\omega_{m, k, \vec{i}}^p (K)} \int\limits_{\partial K} \frac{H_{n-1}(x)^{\frac{p}{n+p}}} {\langle x,N(x)\rangle ^{\frac{n(p-1)}{n+p}}} \ln{\left( \frac{H_{n-1}(x)}{\langle x,N_K(x)\rangle^{n+1}}\right)} d\mu_{\vec{i}} (x)\\ &= - \frac{n}{\omega_{m, k, \vec{i}}^\infty (K)} \int\limits_{\partial K} \frac{H_{n-1}(x)} {\langle x,N_K(x)\rangle ^n} \ln{\left( \frac{H_{n-1}(x)}{\langle x,N_K(x)\rangle^{n+1}}\right)} d\mu_{\vec{i}} (x)\\ &= - \frac{n}{\omega_{m, k, \vec{i}}^\infty (K)} D_{KL}(P_K||Q_K). \end{align*} $$

(ii) We use L’Hospital’s rule

$$ \begin{align*} &\lim\limits_{p \rightarrow 0} \ln \left( \left(\frac{\omega_{m, k, \vec{i}}^p (K^\circ)}{\omega^0_{m, k, \vec{i}}(K^\circ)}\right)^{\frac{n(n+p)}p} \right) = \lim\limits_{p \rightarrow 0} n \frac{\ln \left(\frac{\omega_{m, k, \vec{i}}^p (K^\circ)}{\omega^0_{m, k, \vec{i}}(K^\circ)} \right)}{p(n+p)^{-1}} = \lim\limits_{p \rightarrow 0} (n+p)^2 \frac{\frac{d}{dp}\omega_{m, k, \vec{i}}^p (K^\circ)}{\omega^p_{m, k, \vec{i}}(K^\circ)}\\ &= \lim\limits_{p \rightarrow 0} \frac{(n+p)^2}{\omega^p_{m, k, \vec{i}}(K^\circ)} \int\limits_{\partial K^\circ} \frac{d}{dp} \left( \mathrm{exp} \left( \ln \left( \frac{H_{n-1}(x)^{\frac{p}{n+p}}} {\langle x,N_{K^\circ}(x)\rangle ^{\frac{n(p-1)}{n+p}}} \right) \right) \right) d\mu_{\vec{i}} (x)\\ &= \lim\limits_{p \rightarrow 0} \frac{n}{\omega^p_{m, k, \vec{i}}(K^\circ)} \int\limits_{\partial K^\circ} \frac{H_{n-1}(x)^{\frac{p}{n+p}}} {\langle x,N_{K^\circ}(x)\rangle ^{\frac{n(p-1)}{n+p}}} \ln{\left( \frac{H_{n-1}(x)}{\langle x,N_{K^\circ}(x)\rangle^{n+1}}\right)} d\mu_{\vec{i}} (x)\\ &= \frac{n}{\omega^0_{m, k, \vec{i}}(K^\circ)} \int\limits_{\partial K^\circ} \langle x,N_{K^\circ}(x)\rangle \ln{\left( \frac{H_{n-1}(x)}{\langle x,N_{K^\circ}(x)\rangle^{n+1}}\right)} d\mu_{\vec{i}} (x)\\ &= - \frac{n}{\omega^0_{m, k, \vec{i}}(K^\circ)} D_{KL}(Q_{K^\circ}||P_{K^\circ}).\\[-34pt] \end{align*} $$

4.1 Background on f-divergence

In information theory, probability theory and statistics, an f-divergence is a functional that measures the difference between two (probability) distributions. This notion was introduced by Csiszár [Reference Csiszár12], and independently Morimoto [Reference Morimoto25] and Ali and Silvery [Reference Ali and Silvey2].

Let $(X, \mu )$ be a measure space, and let $P=p \mu $ and $Q=q \mu $ be (probability) measures on X that are absolutely continuous with respect to the measure $\mu $ . Let $f: (0, \infty ) \rightarrow \mathbb {R}$ be a convex or a concave function. The $*$ -adjoint function $f^*:(0, \infty ) \rightarrow \mathbb {R}$ of f is defined by

(4.1) $$ \begin{align} f^*(t) = t f (1/t), \ \ t\in(0, \infty). \end{align} $$

It is obvious that $(f^*)^*=f$ and that $f^*$ is again convex, if f is convex, respectively, concave, if f is concave. Then the f-divergence $D_f(P,Q)$ of the measures P and Q is defined by

(4.2) $$ \begin{align} D_f(P,Q)&= \int_{\{pq>0\} }f\left(\frac{p}{q} \right) q d\mu + f(0)\ Q\left(\{x\in X: p(x) =0\}\right)\nonumber \\ &\quad+ f^*(0) \ P\left(\{x\in X: q(x) =0\}\right), \end{align} $$

provided the expressions exist. Here,

(4.3) $$ \begin{align} f(0) = \lim_{t\downarrow 0} f(t) \ \ \text{ and} \ \ f^*(0) = \lim_{t\downarrow 0} f^*(t). \end{align} $$

We make the convention that $0 \cdot \infty =0$ .

Please note that

(4.4) $$ \begin{align} D_f(P,Q)=D_{f^*}(Q,P). \end{align} $$

With (4.3) and as

$$ \begin{align*}f^*(0) \ P\left(\{x\in X: q(x) =0\}\right) = \int _{\{q=0\}} f^*\left(\frac{q}{p} \right) p d\mu = \int _{\{q=0\}} f\left(\frac{p}{q} \right) q d\mu,\end{align*} $$

we can write in short

(4.5) $$ \begin{align} D_f(P,Q)= \int_{X} f\left(\frac{p}{q} \right) q d\mu. \end{align} $$

Examples of f-divergences are as follows:

1. For $f(t) = t \ln t$ (with $*$ -adjoint function $f^*(t) = - \ln t$ ), the f-divergence is Kullback–Leibler divergence or relative entropy from P to Q (see [Reference Cover and Thomas11])

(4.6) $$ \begin{align} D_{KL}(P\|Q)= \int_{X} p \ln \frac{p}{q} d\mu. \end{align} $$

2. For the convex or concave functions $f(t) = t^\alpha ,$ we obtain the Hellinger integrals (e.g., [Reference Liese and Vajda20])

(4.7) $$ \begin{align} H_\alpha (P,Q) = \int _X p^\alpha q^{1-\alpha} d\mu. \end{align} $$

Those are related to the Rényi divergence of order $\alpha $ , $\alpha \neq 1$ , introduced by Rényi [Reference Rényi28] (for $\alpha>0$ ) as

(4.8) $$ \begin{align} D_\alpha(P\|Q)= \frac{1}{\alpha -1} \ln \left( \int_X p^\alpha q^{1-\alpha} d\mu \right)= \frac{1}{\alpha -1} \ln \left( H_\alpha (P,Q)\right). \end{align} $$

The case $\alpha =1$ is the relative entropy $ D_{KL}(P\|Q)$ .

More on f-divergence can be found in, e.g., [Reference Caglar, Kolesnikov and Werner7–Reference Caglar and Werner9, Reference Liese and Vajda20, Reference Werner37, Reference Werner38, Reference Werner and Ye40].

4.2 f-divergence for the $\mu _{\vec {i}}$ -measure

In [Reference Werner38], f-divergence with respect to the surface area measure $\mu _K$ was introduced for a convex body K in $\mathbb {R}^n$ with $0$ in its interior. We now introduce similarly f-divergence with respect to the measure $\mu _{\vec {i}}$ .

Definition 4.1 Let $f: (0, \infty ) \rightarrow \mathbb {R}$ be a convex or concave function. Let $p_K$ and $q_K$ be as in (3.5). Then the f-divergence $D_f \left (P_K, Q_K \right )$ of a convex body K in $\mathbb {R}^n$ with respect to the $\mu _{\vec {i}}$ -measure is

$$ \begin{align*} D_f \left(P_K, Q_K \right)&= \int_{\partial K } f \left(\frac{p_K}{q_K}\right) q_K d \mu_{\vec{i}}\\ &=\int_{\partial K } f \left( \frac{H_{n-1}}{\langle x,N(x)\rangle^{n+1}} \right) \langle x,N(x)\rangle d \mu_{\vec{i.}} \end{align*} $$

In particular, for $f(t)=t \log t,$ we recover the above Kullback–Leibler divergences of Section 3.4 and for $f(t) = t^{\frac {p}{n+p}}$ , we obtain the weighted $L_p$ -affine surface areas.

Remarks (i) By (4.4),

(4.9) $$ \begin{align} D_f(Q_K,P_K)&= \int _{\partial K} f\left(\frac{q_K}{p_K}\right) p_K d\mu_{\vec{i}} = D_{f^*}(P_K, Q_K) \nonumber \\ &= \int _{\partial K} f^*\left(\frac{p_K}{q_K}\right) q_K d\mu_{\vec{i}} \\ &= \int _{\partial K} f\left(\frac{ {\mathrm{vol}}_n( K^\circ )\langle x, N_K(x) \rangle ^{n+1}}{{\mathrm{vol}}_n(K) H_{n-1} (x)} \right)\frac{H_{n-1}(x)\ d\mathcal{H}^{n-1}} {n\, {\mathrm{vol}}_n(K^\circ) \langle x, N_K(x) \rangle ^{n}}. \nonumber \end{align} $$

(ii) f-divergences can also be expressed as integrals over $S^{n-1}$ ,

(4.10) $$ \begin{align} D_f(P_K, Q_K)&= \int _{S^{n-1}} f\left(\frac{{\mathrm{vol}}_n(K) } {{\mathrm{vol}}_n(K^\circ) f_K(u) h_K(u) ^{n+1}}\right)\frac{ h_K(u)f_K(u)}{n\, {\mathrm{vol}}_n(K)} d\sigma \end{align} $$

and

(4.11) $$ \begin{align} D_f(Q_K,P_K)&= \int _{ S^{n-1}} f\left(\frac{ {\mathrm{vol}}_n(K^\circ) f_K(u) h_K(u) ^{n+1}}{{\mathrm{vol}}_n(K) } \right)\frac{ d\sigma_K} {n\, {\mathrm{vol}}_n(K^\circ )h_K(u)^{n}}. \end{align} $$

(iii) Similar to (4.10) and (4.11), one can define mixed f-divergences for n convex bodies in $\mathbb {R}^n$ . We will not treat those here but will concentrate on f-divergence for one convex body. We also refer to [Reference Ye41], where they have been investigated for functions in $Conv(0,\infty )$ .

Proposition 4.1 Let $f: (0, \infty ) \rightarrow \mathbb {R}$ be a concave function. Then

$$ \begin{align*} D_f \left(P_K, Q_K \right)\leq f\left(\frac{\mu_{\vec{i}}-{\mathrm{vol}}_n(K^{\circ}) }{\mu_{\vec{i}}-{\mathrm{vol}}_n(K) } \right) \, \mu_{\vec{i}}-{\mathrm{vol}}_n(K). \end{align*} $$

If f is convex, the inequality is reversed. If f is linear, equality holds in Proposition 4.1. If f is not linear, equality holds iff K is an ellipsoid.

Proof Let $f: (0, \infty ) \rightarrow \mathbb {R}$ be a concave function. By Jensen’s inequality,

$$ \begin{align*} D_f \left(P_K, Q_K \right) &= \left(\int_{\partial K } f \left( \frac{H_{n-1}}{\langle x,N(x)\rangle^{n+1}} \right) \langle x,N(x)\rangle \, \frac{d \mu_{\vec{i}} }{\mu_{\vec{i}}-{\mathrm{vol}}_n(K)} \right) \, \mu_{\vec{i}}-{\mathrm{vol}}_n(K)\\ &\leq f\left(\int_{\partial K } \frac{H_{n-1}}{\langle x,N(x)\rangle^{n+1}} \langle x,N(x) \rangle \frac{ d \mu_{\vec{i}}} { \mu_{\vec{i}}-{\mathrm{vol}}_n(K)} \right) \, \mu_{\vec{i}}-{\mathrm{vol}}_n(K) \\&= f\left(\frac{\mu_{\vec{i}}-{\mathrm{vol}}_n(K^{\circ} }{\mu_{\vec{i}}-{\mathrm{vol}}_n(K) } \right) \, \mu_{\vec{i}}-{\mathrm{vol}}_n(K). \end{align*} $$

Equality holds in Jensen’s inequality iff either f is linear or $\frac {p_K}{q_K}$ is constant. Indeed, if $f(t) = at +b$ , then

$$ \begin{align*} D_f(P_K, Q_K)&= \int _{\partial K } \left(a \frac{p_K}{q_K} +b \right) q_K \, d \mu_{\vec{i}} = a \int _{\partial K } p_K \, d \mu_{\vec{i}} + b \int _{\partial K } q_K \, d \mu_{\vec{i}} \\ &= a \, \mu_{\vec{i}}-{\mathrm{vol}}_n(K^{\circ}) +b \, \mu_{\vec{i}}-{\mathrm{vol}}_n(K). \end{align*} $$

If f is not linear, equality holds iff $\frac {p_K}{q_K}=c$ , where c is a constant. By the above quoted Theorem 3.3 of Petty, this holds iff K is an ellipsoid. Note that the constant c is different from $0$ , as we assume that K is $C^2_+$ .

Now we show that $D_f({P}_K,{Q}_K)$ are valuations, i.e., for convex bodies K and L such that $K \cup L $ is convex,

(4.12) $$ \begin{align} D_f({P}_{K\cup L},{Q}_{K\cup L}) + D_f({P}_{K\cap L},{Q}_{K\cap L}) = D_f({P}_K,{Q}_K) + D_f({P}_L,{Q}_L). \end{align} $$

Proof To prove (4.12), we proceed as in Schütt [Reference Schütt31]. For completeness, we include the argument. We decompose

$$ \begin{align*}\partial (K \cup L) &= (\partial K \cap \partial L) \cup (\partial K \cap L^c) \cup ( K^c \cap \partial L),\\ \partial (K \cap L) &= (\partial K \cap \partial L) \cup (\partial K \cap \text{int} L) \cup (\text{int} K \cap \partial L),\\ \partial K &= (\partial K \cap \partial L) \cup (\partial K \cap L^c) \cup (\partial K \cap \text{int} L),\\ \partial L &= (\partial K \cap \partial L) \cup (\partial K^c \cap \partial L) \cup (\text{int} K \cap \partial L), \end{align*} $$

where all unions on the right-hand side are disjoint. Note that for x such that the curvatures $\kappa _K(x)$ , $\kappa _L(x)$ , $\kappa _{K\cup L} (x),$ and $\kappa _{K\cap L} (x)$ exist,

(4.13) $$ \begin{align} \langle x, N_K(x)\rangle = \langle x, N_L(x) \rangle = \langle x, N_{K\cap L}(x) \rangle = \langle x, N_{K\cup L}(x) \rangle \end{align} $$

and

(4.14) $$ \begin{align} \kappa_{K\cup L}(x)= \min\{\kappa_K(x), \kappa_L(x)\}, \ \ \kappa_{K\cap L} (x)= \max\{\kappa_K(x), \kappa_L(x)\}. \end{align} $$

To prove (4.12), we split the involved integral using the above decompositions (4.13) and (4.14).