#  Curvature functionals on convex bodies 111Keywords: Steiner formula,
curvature measures, $L_{p}$ Brunn Minkowski theory; 2020 Mathematics Subject
Classification: 52A39 (Primary), 28A75, 52A20, 53A07 (Secondary)

Kateryna Tatarko and Elisabeth M. Werner Partially supported by NSERCPartially
supported by NSF grant DMS-2103482

###### Abstract

We investigate the weighted $L_{p}$ affine surface areas which appear in the
recently established $L_{p}$ Steiner formula of the $L_{p}$ Brunn Minkowski
theory. We show that they are valuations on the set of convex bodies and prove
isoperimetric inequalities for them. We show that they are related to $f$
divergences of the cone measures of the convex body and its polar, namely the
Kullback-Leibler divergence and the Rényi-divergence.

##  1 Introduction

In [34] an $L_{p}$ Steiner formula was proved for the $L_{p}$ affine surface
area, namely, if a convex body $K$ is $C^{2}_{+}$, then we have for all
suitable $t$ and for all $p\in\mathbb{R}$, $p\neq-n$, that

$\displaystyle
as_{p}(K+tB^{n}_{2})=\sum\limits_{k=0}^{\infty}\left[\sum\limits_{m=0}^{k}\binom{\frac{n(1-p)}{n+p}}{{k-m}}\mathcal{W}^{p}_{{m},k}(K)\right]t^{k},$
(1)

where

$as_{p}(K)=\int\limits_{\partial K}\frac{H_{n-1}(x)^{\frac{p}{n+p}}}{\langle
x,N(x)\rangle^{\frac{n(p-1)}{n+p}}}d\mathcal{H}^{n-1}$ (2)

is the $L_{p}$ affine surface area of a convex body $K$, $N(x)$ is the outer
normal to $K$ in $x\in\partial K$, the boundary of $K$, $H_{n-1}(x)$ is the
Gauss curvature in $x$ and $\mathcal{H}^{n-1}$ is the usual surface area
measure on $\partial K$. The Euclidean unit ball centered at $0$ is denoted by
$B^{n}_{2}$.

Identity (1) is the analogue of the classical Steiner formula (e.g., [13, 30])
of the Brunn Minkowski theory in the more recent $L_{p}$ Brunn Minkowski
theory. This theory has as it’s starting point E. Lutwak’s seminal paper [23]
and it has developed immensely since (e.g., [3, 4, 5, 6, 14, 17, 21, 24, 42,
41, 43, 44]). In analogy to the classical theory, the coefficients
$\mathcal{W}^{p}_{{m},k}(K)$ are called $L_{p}$ Steiner coefficients and they
are defined in [34] for a (general) convex body $K$ in $\mathbb{R}^{n}$, for
all $k,m\in\mathbb{N}\cup\\{0\\}$ as

$\displaystyle\hskip-31.29802pt\mathcal{W}^{p}_{m,\,k}(K)=$
$\displaystyle\hskip-31.29802pt\int\limits_{\partial K}\langle
x,N(x)\rangle^{m-k+\frac{n(1-p)}{n+p}}H_{n-1}^{\frac{p}{n+p}}\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq
0\\\
i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=m\end{subarray}}c(n,p,m)\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}H_{j}^{i_{j}}\,d\mathcal{H}^{n-1},$

where the $H_{j}$ are the $j$-th normalized elementary symmetric functions of
the principal curvatures. The $c(n,p,m)$ are certain binomial coefficients,
see [34, 35] for the details. The $L_{p}$ Steiner coefficients were studied in
[35], where it was proved, among other results, that they are valuations on
the set of convex bodies.

###  1.1 Main Results

In this paper, we look at the $L_{p}$ Steiner formula with a different focus.
Expressions also appearing naturally in formula (1) are weighted $L_{p}$
affine surface areas, $\mu_{\vec{i}}-as_{p}(K)$, which we define in Section
2.2. We investigate in detail those weighted $L_{p}$ affine surface areas in
Section 3. We show that they are homogeneous of a certain degree and are
invariant under rotations and reflections. We show that they are valuations on
the set of convex bodies. Valuations have become a vitally import subject of
study in convexity and affine and differential geometry, e.g., [1, 8, 15, 16,
22, 33]. The weighted $L_{p}$ affine surface areas satisfy isoperimetric
inequalities which generalize the $L_{p}$ affine isoperimetric inequalities of
[23, 39]. This is shown in Theorem 3.2. Theorem 3.2 Let
$s\neq-n,r\neq-n,t\neq-n$ be real numbers. Let $K$ be a $C^{2}_{+}$ convex
body in $\mathbb{R}^{n}$ with centroid at the origin. (i) If
$\frac{(n+r)(t-s)}{(n+t)(r-s)}>1$, then

$\mu_{\vec{i}}-as_{r}(K)\leq\big{(}\mu_{\vec{i}}-as_{t}(K)\big{)}^{\frac{(r-s)(n+t)}{(t-s)(n+r)}}\big{(}\mu_{\vec{i}}-as_{s}(K)\big{)}^{\frac{(t-r)(n+s)}{(t-s)(n+r)}}.$

(ii) If $\frac{(n+r)t}{(n+t)r}>1$, then

$\frac{\mu_{\vec{i}}-as_{r}(K)}{\mu_{\vec{i}}-{\rm vol}_{n}(K)}\leq
n^{\frac{n(t-r)}{t(n+r)}}\bigg{(}\frac{\mu_{\vec{i}}-as_{t}(K)}{\mu_{\vec{i}}-{\rm
vol}_{n}(K)}\bigg{)}^{\frac{r(n+t)}{{t(n+r)}}}.$

Equality holds in the above inequalities, if and only if $K$ is an ellipsoid.
We show in Section 3.3 that the weighted $L_{p}$ affine surface areas have
natural geometric interpretations in terms of certain convex bodies associated
with the given convex body $K$. We prove a monotonicity behavior in the
parameter $p$ for the weighted $L_{p}$ affine surface areas which allows to
establish asymptotics for the weighted $L_{p}$ affine surface areas. These
asymptotics connects them to entropy powers, namely to the Kullbak-Leibler
divergence $D_{KL}$ of the cone measures of $K$ and its polar $K^{\circ}$,
$Q_{K}$ and $P_{K}$. We quote the relevant Theorem 3.8 and refer to Section
3.4 for the details. We put

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

and then the following theorem holds. Theorem 3.8 Let $K$ be a $C^{2}_{+}$
convex body in $\mathbb{R}^{n}$ with centroid at the origin. Then (i)

$\lim\limits_{p\rightarrow\infty}\left(\frac{\omega_{m,k,\vec{i}}^{p}(K)}{\omega^{\infty}_{m,k,\vec{i}}(K)}\right)^{n+p}=\textup{exp}\left(-\frac{n\,D_{KL}(P_{K}||Q_{K})}{\mu_{\vec{i}}-{\rm
vol}_{n}(K^{\circ})}\right).$

(ii)

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

This leads naturally to consider more general $f$-divergences than just the
Kullbak-Leibler divergence. We treat that in Section 4, where we also observe
that the weighted $L_{p}$ affine surface areas themselves are special
$f$-divergences. Throughout the paper we assume that the convex bodies $K$ are
$C^{2}_{+}$, i.e., $K$ has twice continuously differentiable boundary with
strictly positive Gauss curvature everywhere and such that $0$ is the centroid
of $K$, $0=\frac{1}{{\rm vol}_{n}(K)}\int_{K}xdx$.

##  2 Weighted $L_{p}$-affine surface areas

###  2.1 Background from differential geometry

For more information and the details in this section we refer to e.g., [13,
30]. 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 $j$-th normalized elementary symmetric functions of the principal
curvatures are denoted by $H_{j}$. They are defined as follows

$H_{j}={n-1\choose j}^{-1}\sum_{1\leq i_{1}<\dots<i_{j}\leq
n-1}k_{i_{1}}\cdots k_{i_{j}}$ (3)

for $j=1,\dots,n-1$ and $H_{0}=1.$ Note that

$H_{1}=\frac{1}{n-1}\sum_{1\leq i\leq n-1}k_{i}$

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

$H_{n-1}=\prod_{i=1}^{n-1}k_{i}$

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 $H_{n-1}\neq 0.$ 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 $j$-th 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

$s_{j}={n-1\choose j}^{-1}\sum_{1\leq i_{1}<\dots<i_{j}\leq
n-1}r_{i_{1}}\cdots r_{i_{j}}.$ (4)

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

$r_{i}(u)=\frac{1}{k_{i}(\bar{\xi}_{K}(u))}.$

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

$s_{j}={n-1\choose j}^{-1}\sum_{1\leq i_{1}<\dots<i_{j}\leq
n-1}\frac{1}{k_{i_{1}}(\bar{\xi}_{K}(u))\cdots
k_{i_{j}}(\bar{\xi}_{K}(u))}=\frac{H_{n-1-j}}{H_{n-1}}\Big{(}\bar{\xi}_{K}(u)\Big{)}$

and

$H_{j}=\frac{s_{n-1-j}}{s_{n-1}}\Big{(}\nu(x)\Big{)},$

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

###  2.2 Definitions

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
$i_{1}+2i_{2}+\cdots(n-1)i_{n-1}=m$ and all $p\in\mathbb{R}$, $p\neq-n$, we
define

$\omega_{m,k,\vec{i}}^{p}(K)=\int\limits_{\partial
K}\frac{H_{n-1}(x)^{\frac{p}{n+p}}}{\langle
x,N(x)\rangle^{\frac{n(p-1)}{n+p}}}\,\langle
x,N(x)\rangle^{m-k}\,\prod\limits_{j=1}^{n-1}\left\\{{n-1\choose
j}^{i_{j}}H_{j}^{i_{j}}(x)\right\\}\,d\mathcal{H}^{n-1}(x)$ (5)

and

$u_{m,k,\vec{i}}^{p}(K)=\int\limits_{S^{n-1}}\frac{f_{K}(u)^{\frac{n}{n+p}}}{h_{K}(u)^{\frac{n(p-1)}{n+p}}}\,h_{K}(u)^{m-k}\,\frac{\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}s_{n-1-j}^{i_{j}}(u)}{s_{n-1}^{\sum\limits_{j}i_{j}+\frac{p}{n+p}}(u)}\,d\mathcal{H}^{n-1}(u).$
(6)

###### 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

$\omega_{m,k,\vec{i}}^{p}(K)=u_{m,k,\vec{i}}^{p}(K).$ (7)

Denote

$c(n)=\prod\limits_{j=1}^{n-1}{n-1\choose j}^{i_{j}}.$

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

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

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

$d\nu_{m,k,\vec{i}}(u)=c(n)\,h^{m-k}_{K}(u)\frac{\prod\limits_{j=1}^{n-1}s_{n-1-j}^{i_{j}}(u)}{s_{n-1}^{\sum\limits_{j}i_{j}+\frac{p}{n+p}}(u)}d\mathcal{H}^{n-1}(u).$

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

$\mu_{\vec{i}}-as_{p}(K)=\omega_{m,k,\vec{i}}^{p}(K)=\int\limits_{\partial
K}\frac{H_{n-1}(x)^{\frac{p}{n+p}}}{\langle
x,N(x)\rangle^{\frac{n(p-1)}{n+p}}}d\mu_{\vec{i}}(x).$ (8)

Definition (2) explains that those can be considered as $L_{p}$-affine surface
area weighted by the measure $\mu_{\vec{i}}$. In particular,

$\omega_{m,k,\vec{i}}^{0}(K)=\int\limits_{\partial K}\langle x,N(x)\rangle
d\mu_{\vec{i}}(x)=n(\mu_{\vec{i}}-{\rm vol}_{n}(K))$

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

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

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

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

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 [23]. Then

$u_{m,k,\vec{i}}^{p}(K)=c(n)\int\limits_{S^{n-1}}f_{p}(K,u)h_{K}(u)^{m-k}\,\frac{\prod\limits_{j=1}^{n-1}s_{n-1-j}^{i_{j}}(u)}{s_{n-1}^{\sum\limits_{j}i_{j}+\frac{p}{n+p}}(u)}\,d\mathcal{H}^{n-1}(u).$

###  2.3 Special cases

Note that

$\omega_{0,0,0}^{p}(K)=u_{0,0,0}^{p}(K)=as_{p}(K),$

where we denote the sequence $\vec{i}=\\{0,\dots,0\\}$ as $0$. 1\. When $k=m$,
we get

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

2\. $m=0$ implies that $\vec{i}=0$ and (5) simplifies to

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

(see [34, Eq. (29)]).

3\. For the Euclidean unit ball $B^{n}_{2}$, we get

$\displaystyle\omega_{m,k,\vec{i}}^{p}(B^{n}_{2})={\rm
vol}_{n-1}(B^{n}_{2})\prod\limits_{j=1}^{n-1}{n-1\choose j}^{i_{j}}=c(n)\,{\rm
vol}_{n-1}(\partial B^{n}_{2})$ (9)

which does not depend on $k$. Note that if $i_{0}\neq 0$ and
$i_{1}=\dots=i_{n-1}=0$, that is $\vec{i}=\\{i_{0},0,\dots,0\\}$, then
$\omega_{m,k,\vec{i}}^{p}(B^{n}_{2})={\rm vol}_{n-1}(\partial B^{n}_{2})$.

4\. If $p=0$, we have

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

When in addition $k=m$, we get

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

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

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

##  3 Properties of the weighted $L_{p}$-affine surface areas

###  3.1 Valuation, Invariance and Homogeneity

###### Proposition 3.1

Let $K$ be a $C^{2}_{+}$ convex body in $\mathbb{R}^{n}$ with centroid at the
origin. Let $p\in\mathbb{R}$, $p\neq-n$. Then $\omega_{m,k,\vec{i}}^{p}(K)$ is
a $n\frac{n-p}{n+p}-k$-homogeneous valuation that is invariant under rotations
and reflections.

Proof  The proof follows immediately from results in [35]. We present an
outline of the proof for completeness. 1\. Valuation. As was shown in [35,
Theorem 5.9], for all $1\leq i_{1},\dots,i_{n-1}\leq n-1$, $1\leq j\leq n-1$,
and $\alpha_{1},\dots,\alpha_{j}\geq 0$,

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

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 [35, 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 [35, 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 $1\leq
j\leq n-1$,

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

Thus, using these observations and [35, Proposition 5.4], we get

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

###  3.2 Inequalities

###### Theorem 3.2

Let $s\neq-n,r\neq-n,t\neq-n$ be real numbers. Let $K$ be a $C^{2}_{+}$ convex
body in $\mathbb{R}^{n}$ with centroid at the origin. (i) If
$\frac{(n+r)(t-s)}{(n+t)(r-s)}>1$, then

$\mu_{\vec{i}}-as_{r}(K)\leq\big{(}\mu_{\vec{i}}-as_{t}(K)\big{)}^{\frac{(r-s)(n+t)}{(t-s)(n+r)}}\big{(}\mu_{\vec{i}}-as_{s}(K)\big{)}^{\frac{(t-r)(n+s)}{(t-s)(n+r)}}.$

(ii) If $\frac{(n+r)t}{(n+t)r}>1$, then

$\frac{\mu_{\vec{i}}-as_{r}(K)}{\mu_{\vec{i}}-{\rm vol}_{n}(K)}\leq
n^{\frac{n(t-r)}{t(n+r)}}\bigg{(}\frac{\mu_{\vec{i}}-as_{t}(K)}{\mu_{\vec{i}}-{\rm
vol}_{n}(K)}\bigg{)}^{\frac{r(n+t)}{{t(n+r)}}}.$

Equality holds in the above inequalities, if and only if $K$ is an ellipsoid.

Proof

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

$\displaystyle\mu_{\vec{i}}-as_{r}(K)=\omega_{m,k,\vec{i}}^{r}(K)$
$\displaystyle=$ $\displaystyle\int\limits_{\partial
K}\frac{H_{n-1}(x)^{\frac{r}{n+r}}}{\langle
x,N(x)\rangle^{\frac{n(r-1)}{n+r}}}\,\langle
x,N(x)\rangle^{m-k}\,\prod\limits_{j=1}^{n-1}\left\\{{n-1\choose
j}^{i_{j}}H_{j}^{i_{j}}(x)\right\\}\,d\mathcal{H}^{n-1}(x)$ $\displaystyle=$
$\displaystyle\int_{\partial K}\left(\frac{H_{n-1}(x)^{\frac{t}{n+t}}}{\langle
x,N(x)\rangle^{\frac{n(t-1)}{n+t}}}\right)^{\frac{(r-s)(n+t)}{(t-s)(n+r)}}\left(\frac{H_{n-1}(x)^{\frac{s}{n+s}}}{\langle
x,N(x)\rangle^{\frac{n(s-1)}{n+s}}}\right)^{\frac{(t-r)(n+s)}{(t-s)(n+r)}}\,d\mu_{\vec{i}}(x)$
$\displaystyle\leq$
$\displaystyle\big{(}\omega_{m,k,\vec{i}}^{t}(K)\big{)}^{\frac{(r-s)(n+t)}{(t-s)(n+r)}}\big{(}\omega_{m,k,\vec{i}}^{s}(K)\big{)}^{\frac{(t-r)(n+s)}{(t-s)(n+r)}}$
$\displaystyle=$
$\displaystyle\big{(}\mu_{\vec{i}}-as_{t}(K)\big{)}^{\frac{(r-s)(n+t)}{(t-s)(n+r)}}\big{(}\mu_{\vec{i}}-as_{s}(K)\big{)}^{\frac{(t-r)(n+s)}{(t-s)(n+r)}}.$

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

$\displaystyle\mu_{\vec{i}}-as_{r}(K)=\omega_{m,k,\vec{i}}^{r}(K)$
$\displaystyle=$ $\displaystyle\int\limits_{\partial
K}\frac{H_{n-1}(x)^{\frac{r}{n+r}}}{\langle
x,N(x)\rangle^{\frac{n(r-1)}{n+r}}}\,d\mu_{\vec{i}}(x)=\int\limits_{\partial
K}\left(\frac{H_{n-1}(x)^{\frac{t}{n+t}}}{\langle
x,N(x)\rangle^{\frac{n(t-1)}{n+t}}}\right)^{\frac{r(n+t)}{t(n+r)}}\frac{d\mu_{\vec{i}}(x)}{\langle
x,N(x)\rangle^{\frac{(r-t)n}{(n+r)t}}}$ $\displaystyle\leq$
$\displaystyle\big{(}\omega_{m,k,\vec{i}}^{t}(K)\big{)}^{\frac{r(n+t)}{t(n+r)}}\big{(}\omega_{m,k,\vec{i}}^{0}(K)\big{)}^{\frac{(t-r)n}{(n+r)t}}$
$\displaystyle=$
$\displaystyle\bigg{(}\mu_{\vec{i}}-as_{t}(K)\bigg{)}^{\frac{r(n+t)}{{t(n+r)}}}\bigg{(}n(\mu_{\vec{i}}-{\rm
vol}_{n}(K))\bigg{)}^{\frac{n(t-r)}{{t(n+r)}}}.$

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

$\frac{H_{n-1}}{\langle x,N(x)\rangle^{n+1}}=\textup{constant}$

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

$\frac{H_{n-1}}{\langle x,N(x)\rangle^{n+1}}=\textup{constant}$

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

######  Theorem 3.3 ([27])

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 $\partial K$

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

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

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

Equality holds if and only if $K$ is a ball.

Proof

$\displaystyle\omega_{m,s,\vec{i}}^{p}(K)=\int\limits_{\partial
K}\frac{H_{n-1}(x)^{\frac{p}{n+p}}}{\langle
x,N(x)\rangle^{\frac{n(p-1)}{n+p}}}\,\langle
x,N(x)\rangle^{m-s}\,\prod\limits_{j=1}^{n-1}\left\\{{n-1\choose
j}^{i_{j}}H_{j}^{i_{j}}(x)\right\\}\,d\mathcal{H}^{n-1}(x)$ $\displaystyle=$
$\displaystyle\int\limits_{\partial
K}\left(\frac{H_{n-1}(x)^{\frac{p}{n+p}}}{\langle
x,N(x)\rangle^{\frac{n(p-1)}{n+p}}}\langle
x,N(x)\rangle^{m-k}c(n)\prod\limits_{j=1}^{n-1}H_{j}^{i_{j}}\right)^{\frac{s-r}{k-r}}\left(\frac{H_{n-1}(x)^{\frac{p}{n+p}}}{\langle
x,N(x)\rangle^{\frac{n(p-1)}{n+p}}}\langle
x,N(x)\rangle^{m-r}c(n)\prod\limits_{j=1}^{n-1}H_{j}^{i_{j}}\right)^{\frac{k-s}{k-r}}d\mathcal{H}^{n-1}$
$\displaystyle\leq$
$\displaystyle\big{(}\omega_{m,k,\vec{i}}^{p}(K)\big{)}^{\frac{s-r}{k-r}}\big{(}\omega_{m,r,\vec{i}}^{p}(K)\big{)}^{\frac{k-s}{k-r}}.$

###### Remark 3.5

When $\vec{i}=0$ and $k=0$, we recover $L_{p}$ affine isoperimetric
inequalities of [39].

We also obtain monotonicity behaviors.

###### Corollary 3.6

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

(i) The function
$p\rightarrow\left(\frac{\omega_{m,k,\vec{i}}^{p}(K)}{\omega^{0}_{m,k,\vec{i}}(K)}\right)^{\frac{n+p}{p}}$
is increasing in $p\in(-n,\infty)$ and $p\in(-\infty,-n)$.

(i) The function
$p\rightarrow\left(\frac{\omega_{m,k,\vec{i}}^{p}(K)}{\omega^{0}_{m,k,\vec{i}}(K)}\right)^{n+p}$
is decreasing in $p\in(-n,\infty)$ and $p\in(-\infty,-n)$.

(iii) The inequalities are strict unless $K$ is an ellipsoid.

Proof. The statement (i) follows immediately from Theorem 3.2 (ii). The
statement (ii) follows from Theorem 3.2 (i), by letting $s\rightarrow\infty$.
The statement (iii) follows from the equality characterizations.

###  3.3 Geometric interpretations

We recall several constructions of convex bodies associated with a given
convex body $K$. Namely, 1\. Weighted floating bodies [36] Let $K$ be a convex
body in ${\mathbb{R}}^{n}$ and denote by $\lambda$ the Lebesgue measue 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 [36] (see also [3, 4, 5]) 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 $K$,

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

It was shown in [36] that

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

2\. Surface bodies [32] Let $K$ be a convex body and $f:\partial
K\rightarrow\mathbb{R}$ 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 [32] 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 $\partial K$,

$S(K,f,s)=\bigcap_{\mathbb{P}_{f}(\partial K\cap H^{-})\leq s}H^{+}.$

It was shown in [32] that

$2\bigg{(}\mbox{vol}_{n-1}(B^{n-1}_{2})\bigg{)}^{\frac{2}{n-1}}\lim_{s\to
0}\frac{\mbox{\rm vol}_{n}(K)-\mbox{\rm
vol}_{n}(S(K,f,s))}{s^{\frac{2}{n-1}}}=\int_{\partial
K}\frac{H_{n-1}^{\frac{1}{n-1}}}{f^{\frac{2}{n-1}}}\,d\mathcal{H}^{n-1}.$

3\. Random polytopes [31] A random polytope is the convex hull of finitely
many points that are chosen with respect to a probability measure. In [31],
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

$\mathbb{E}(f,N)=\mathbb{E}(\mathbb{P}_{f},N)=\int_{\partial
K}\cdots\int_{\partial
K}\mbox{vol}_{n}([x_{1},\dots,x_{N}])d\mathbb{P}_{f}(x_{1})\dots
d\mathbb{P}_{f}(x_{N}),$

where $[x_{1},\dots,x_{N}]$ is the convex hull of the points
$x_{1},\dots,x_{N}$. It was shown in [31] that under mild smoothness
assumptions on $K$,

$\lim_{N\to\infty}\frac{\mbox{\rm
vol}_{n}(K)-\mathbb{E}(f,N)}{\left(\frac{1}{N}\right)^{\frac{2}{n-1}}}=c_{n}\int_{\partial
K}\frac{H_{n-1}^{\frac{1}{n-1}}}{f^{\frac{2}{n-1}}}\,d\mathcal{H}^{n-1}.$

$c_{n}=\frac{(n-1)^{\frac{n+1}{n-1}}\Gamma\left(n+1+\tfrac{2}{n-1}\right)}{2(n+1)!(\mbox{\rm
vol}_{n-2}(\partial B_{2}^{n-1}))^{\frac{2}{n-1}}}$

4\. Ulam floating bodies [19] Given a Borel measure $\mu$ on $\mathbb{R}^{n}$,
the metronoid associated to $\mu$ was introduced by Huang and Slomka [18] and
is the convex set defined by

$\mathcal{M}(\mu)=\bigcup_{\begin{subarray}{c}0\leq f\leq 1,\\\
\int_{\mathbb{R}^{n}}fd{\mu}=1\end{subarray}}\left\\{\int_{\mathbb{R}^{n}}yf\left(y\right)d{\mu}\left(y\right)\right\\},$

where the union is taken over all functions $0\leq f\leq 1$ for which
$\int_{\mathbb{R}^{n}}fd{\mu=1}$ 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 $\mathbbm{1}_{K}$ be
the characteristic function of $K$, and $\mu$ the measure whose density with
respect to Lebesgue measure is $\delta^{-1}\mathbbm{1}_{K}$. Then
$\mathcal{M}_{\delta}(K):=\mathcal{M}(\mu)$. In [19] weighted variations of
$\mathcal{M}_{\delta}(K)$ were defined where the weight is given by a positive
continuous function $f:K\to\mathbb{R}$,

$\mathcal{M}_{\delta}(K,f):=\mathcal{M}\left(\frac{f\left(x\right)}{\delta}\mathbbm{1}_{K}\left(x\right)dx\right).$

It was shown in [19] that

$\lim_{\delta\searrow
0}\frac{\mbox{vol}_{n}\left(K\right)-\mbox{vol}_{n}\left(\mathcal{M}_{\delta}(K,f)\right)}{\delta^{\frac{2}{n+1}}}=2\frac{n+1}{n+3}\left(\frac{{\rm
vol}_{n-1}(B_{2}^{n-1})}{n+1}\right)^{\frac{2}{n+1}}\int_{\partial
K}\frac{H_{n-1}^{\frac{1}{n+1}}}{f^{\frac{2}{n+1}}}\,d\mathcal{H}^{n-1}.$

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

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

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

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

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

$K_{f,s}=F(K,f,s),\hskip 5.69054pt\text{or}\hskip
5.69054ptK_{f,s}=S(K,f,s),\hskip 5.69054pt\text{or}\hskip
5.69054ptK_{f,s}=\mathcal{M}_{s}(K,f),\hskip 5.69054pt\text{or}\hskip
5.69054pt{\rm vol}_{n}(K_{f,s})=\mathbb{E}(f,N),$

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 $p\neq-n$ be a real number.

$c_{n}\lim_{s\to 0}\frac{\mbox{\rm vol}_{n}(K)-\mbox{\rm
vol}_{n}(K_{f,s})}{s^{\frac{2}{n-1}}}=\omega_{m,k,\vec{i}}^{p}(K).$

###  3.4 Asymptotics for the weighted $L_{p}$-affine surface areas

Let $(X,\mu)$ be a measure space and $dP=pd\mu$ and $Q=qd\mu$ 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 [11]

$D_{KL}(P||Q)=\int\limits_{X}p\log{\frac{p}{q}}d\mu.$ (10)

The information inequality (also called Gibb’s inequality) [11] holds for the
Kullback-Leibler divergence. Let $P$ and $Q$ be as above, then

$D_{KL}(P||Q)\geq 0.$ (11)

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

$p_{K}(x)=\frac{H_{n-1}}{\langle x,N(x)\rangle^{n}},\qquad q_{K}(x)=\langle
x,N(x)\rangle.$ (12)

We let

$P_{K}=\frac{H_{n-1}}{\langle x,N(x)\rangle^{n}}\mu_{\vec{i}},\qquad
Q_{K}=\langle x,N(x)\rangle\mu_{\vec{i}}.$ (13)

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

$cm_{K}(A)={\rm vol}_{n}(\\{ta:\ a\in A,\,t\in[0,1]\\}).$

It is well-known (see, e.g, [26] 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 $K^{\circ}$,

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

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
$K$ $N_{K}:\partial K\rightarrow S^{n-1},\ x\longmapsto N_{K}(x)$,

$N_{K}^{-1}N_{K^{\circ}}cm_{K^{\circ}}$

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

$\mu_{\vec{i}}-cm_{K}=\frac{1}{n}\int\limits_{A}\langle
x,N(x)\rangle\mu_{\vec{i}}(x),$ (14)
$\mu_{\vec{i}}-cm_{K^{\circ}}(A)=\frac{1}{n}\int\limits_{A}\frac{H_{n-1}}{\langle
x,N(x)\rangle^{n}}\mu_{\vec{i}}(x).$ (15)

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)

$\lim\limits_{p\rightarrow\infty}\left(\frac{\omega_{m,k,\vec{i}}^{p}(K)}{\omega^{\infty}_{m,k,\vec{i}}(K)}\right)^{n+p}=\textup{exp}\left(-\frac{n\,D_{KL}(P_{K}||Q_{K})}{\mu_{\vec{i}}-{\rm
vol}_{n}(K^{\circ})}\right).$

(ii)

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

Proof. (i) We use l’Hospitals rule

$\displaystyle\lim\limits_{p\rightarrow\infty}\ln\left(\left(\frac{\omega_{m,k,\vec{i}}^{p}(K)}{\omega^{\infty}_{m,k,\vec{i}}(K)}\right)^{n+p}\right)=\lim\limits_{p\rightarrow\infty}\frac{\ln\left(\frac{\omega_{m,k,\vec{i}}^{p}(K)}{\omega^{\infty}_{m,k,\vec{i}}(K)}\right)}{(n+p)^{-1}}=-\lim\limits_{p\rightarrow\infty}(n+p)^{2}\frac{\frac{d}{dp}\left(\omega_{m,k,\vec{i}}^{p}(K)\right)}{\omega_{m,k,\vec{i}}^{p}(K)}$
$\displaystyle=-\lim\limits_{p\rightarrow\infty}\frac{(n+p)^{2}}{\omega_{m,k,\vec{i}}^{p}(K)}\int\limits_{\partial
K}\frac{d}{dp}\left(\textup{exp}\left(\ln\left(\frac{H_{n-1}(x)^{\frac{p}{n+p}}}{\langle
x,N_{K}(x)\rangle^{\frac{n(p-1)}{n+p}}}\right)\right)\right)d\mu_{\vec{i}}(x)$
$\displaystyle=-\lim\limits_{p\rightarrow\infty}\frac{n}{\omega_{m,k,\vec{i}}^{p}(K)}\int\limits_{\partial
K}\frac{H_{n-1}(x)^{\frac{p}{n+p}}}{\langle
x,N(x)\rangle^{\frac{n(p-1)}{n+p}}}\ln{\left(\frac{H_{n-1}(x)}{\langle
x,N_{K}(x)\rangle^{n+1}}\right)}d\mu_{\vec{i}}(x)$
$\displaystyle=-\frac{n}{\omega_{m,k,\vec{i}}^{\infty}(K)}\int\limits_{\partial
K}\frac{H_{n-1}(x)}{\langle
x,N_{K}(x)\rangle^{n}}\ln{\left(\frac{H_{n-1}(x)}{\langle
x,N_{K}(x)\rangle^{n+1}}\right)}d\mu_{\vec{i}}(x)$
$\displaystyle=-\frac{n}{\omega_{m,k,\vec{i}}^{\infty}(K)}D_{KL}(P_{K}||Q_{K}).$

(ii) We use l’Hospitals rule

$\displaystyle\lim\limits_{p\rightarrow
0}\ln\left(\left(\frac{\omega_{m,k,\vec{i}}^{p}(K^{\circ})}{\omega^{0}_{m,k,\vec{i}}(K^{\circ})}\right)^{\frac{n(n+p)}{p}}\right)=\lim\limits_{p\rightarrow
0}n\frac{\ln\left(\frac{\omega_{m,k,\vec{i}}^{p}(K^{\circ})}{\omega^{0}_{m,k,\vec{i}}(K^{\circ})}\right)}{p(n+p)^{-1}}=\lim\limits_{p\rightarrow
0}(n+p)^{2}\frac{\frac{d}{dp}\omega_{m,k,\vec{i}}^{p}(K^{\circ})}{\omega^{p}_{m,k,\vec{i}}(K^{\circ})}$
$\displaystyle=\lim\limits_{p\rightarrow
0}\frac{(n+p)^{2}}{\omega^{p}_{m,k,\vec{i}}(K^{\circ})}\int\limits_{\partial
K^{\circ}}\frac{d}{dp}\left(\textup{exp}\left(\ln\left(\frac{H_{n-1}(x)^{\frac{p}{n+p}}}{\langle
x,N_{K^{\circ}}(x)\rangle^{\frac{n(p-1)}{n+p}}}\right)\right)\right)d\mu_{\vec{i}}(x)$
$\displaystyle=\lim\limits_{p\rightarrow
0}\frac{n}{\omega^{p}_{m,k,\vec{i}}(K^{\circ})}\int\limits_{\partial
K^{\circ}}\frac{H_{n-1}(x)^{\frac{p}{n+p}}}{\langle
x,N_{K^{\circ}}(x)\rangle^{\frac{n(p-1)}{n+p}}}\ln{\left(\frac{H_{n-1}(x)}{\langle
x,N_{K^{\circ}}(x)\rangle^{n+1}}\right)}d\mu_{\vec{i}}(x)$
$\displaystyle=\frac{n}{\omega^{0}_{m,k,\vec{i}}(K^{\circ})}\int\limits_{\partial
K^{\circ}}\langle x,N_{K^{\circ}}(x)\rangle\ln{\left(\frac{H_{n-1}(x)}{\langle
x,N_{K^{\circ}}(x)\rangle^{n+1}}\right)}d\mu_{\vec{i}}(x)$
$\displaystyle=-\frac{n}{\omega^{0}_{m,k,\vec{i}}(K^{\circ})}D_{KL}(Q_{K^{\circ}}||P_{K^{\circ}}).$

##  4  $f$-divergence

The results in the previous section lead naturally to consider more general
$f$-divergences than just the Kullback-Leibler divergence.

###  4.1 Background on $f$-divergence.

In information theory, probability theory and statistics, an $f$-divergence is
a function that measures the difference between two (probability)
distributions. This notion was introduced by Csiszár [12], and independently
Morimoto [25] and Ali & Silvery [2].

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

$f^{*}(t)=tf(1/t),\ \ t\in(0,\infty).$ (16)

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

$\displaystyle D_{f}(P,Q)$ $\displaystyle=$
$\displaystyle\int_{\\{pq>0\\}}f\left(\frac{p}{q}\right)qd\mu+f(0)\
Q\left(\\{x\in X:p(x)=0\\}\right)$ (17) $\displaystyle+$ $\displaystyle
f^{*}(0)\ P\left(\\{x\in X:q(x)=0\\}\right),$

provided the expressions exist. Here

$f(0)=\lim_{t\downarrow 0}f(t)\ \ \text{ and}\ \ f^{*}(0)=\lim_{t\downarrow
0}f^{*}(t).$ (18)

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

$D_{f}(P,Q)=D_{f^{*}}(Q,P).$ (19)

With (18) and as

$f^{*}(0)\ P\left(\\{x\in
X:q(x)=0\\}\right)=\int_{\\{q=0\\}}f^{*}\left(\frac{q}{p}\right)pd\mu=\int_{\\{q=0\\}}f\left(\frac{p}{q}\right)qd\mu,$

we can write in short

$D_{f}(P,Q)=\int_{X}f\left(\frac{p}{q}\right)qd\mu.$ (20)

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 [11])

$D_{KL}(P\|Q)=\int_{X}p\ln\frac{p}{q}d\mu.$ (21)

2\. For the convex or concave functions $f(t)=t^{\alpha}$ we obtain the
Hellinger integrals (e.g. [20])

$H_{\alpha}(P,Q)=\int_{X}p^{\alpha}q^{1-\alpha}d\mu.$ (22)

Those are related to the Rényi divergence of order $\alpha$, $\alpha\neq 1$,
introduced by Rényi [28] (for $\alpha>0$) as

$D_{\alpha}(P\|Q)=\frac{1}{\alpha-1}\ln\left(\int_{X}p^{\alpha}q^{1-\alpha}d\mu\right)=\frac{1}{\alpha-1}\ln\left(H_{\alpha}(P,Q)\right).$
(23)

The case $\alpha=1$ is the relative entropy $D_{KL}(P\|Q)$. More on
$f$-divergence can be found in e.g. [7, 9, 10, 20, 37, 38, 40].

###  4.2 $f$-divergence for the $\mu_{\vec{i}}$-measure

In [38], $f$-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 (12). 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

$\displaystyle D_{f}\left(P_{K},Q_{K}\right)$ $\displaystyle=$
$\displaystyle\int_{\partial
K}f\left(\frac{p_{K}}{q_{K}}\right)q_{K}d\mu_{\vec{i}}$ $\displaystyle=$
$\displaystyle\int_{\partial K}f\left(\frac{H_{n-1}}{\langle
x,N(x)\rangle^{n+1}}\right)\langle x,N(x)\rangle d\mu_{\vec{i}}$

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 (19),

$\displaystyle D_{f}(Q_{K},P_{K})$ $\displaystyle=$
$\displaystyle\int_{\partial
K}f\left(\frac{q_{K}}{p_{K}}\right)p_{K}d\mu_{\vec{i}}=D_{f^{*}}(P_{K},Q_{K})$
(24) $\displaystyle=$ $\displaystyle\int_{\partial
K}f^{*}\left(\frac{p_{K}}{q_{K}}\right)q_{K}d\mu_{\vec{i}}$ $\displaystyle=$
$\displaystyle\int_{\partial K}f\left(\frac{{\rm vol}_{n}(K^{\circ})\langle
x,N_{K}(x)\rangle^{n+1}}{{\rm vol}_{n}(K)H_{n-1}(x)}\right)\frac{H_{n-1}(x)\
d\mathcal{H}^{n-1}}{n\,{\rm vol}_{n}(K^{\circ})\langle
x,N_{K}(x)\rangle^{n}}.$

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

$\displaystyle D_{f}(P_{K},Q_{K})$ $\displaystyle=$
$\displaystyle\int_{S^{n-1}}f\left(\frac{{\rm vol}_{n}(K)}{{\rm
vol}_{n}(K^{\circ})f_{K}(u)h_{K}(u)^{n+1}}\right)\frac{h_{K}(u)f_{K}(u)}{n\,{\rm
vol}_{n}(K)}d\sigma$ (25)

and

$\displaystyle D_{f}(Q_{K},P_{K})$ $\displaystyle=$
$\displaystyle\int_{S^{n-1}}f\left(\frac{{\rm
vol}_{n}(K^{\circ})f_{K}(u)h_{K}(u)^{n+1}}{{\rm
vol}_{n}(K)}\right)\frac{d\sigma_{K}}{n\,{\rm
vol}_{n}(K^{\circ})h_{K}(u)^{n}}.$ (26)

(iii) Similar to (25) and (26), 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 [41],
where they have been investigated for functions in $Conv(0,\infty)$.

###### Proposition 4.2

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

$\displaystyle D_{f}\left(P_{K},Q_{K}\right)\leq
f\left(\frac{\mu_{\vec{i}}-{\rm vol}_{n}(K^{\circ})}{\mu_{\vec{i}}-{\rm
vol}_{n}(K)}\right)\,\mu_{\vec{i}}-{\rm vol}_{n}(K).$

If $f$ is convex, the inequality is reversed. If $f$ is linear, equality holds
in (4.2). 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

$\displaystyle D_{f}\left(P_{K},Q_{K}\right)$ $\displaystyle=$
$\displaystyle\left(\int_{\partial K}f\left(\frac{H_{n-1}}{\langle
x,N(x)\rangle^{n+1}}\right)\langle
x,N(x)\rangle\,\frac{d\mu_{\vec{i}}}{\mu_{\vec{i}}-{\rm
vol}_{n}(K)}\right)\,\mu_{\vec{i}}-{\rm vol}_{n}(K)$ $\displaystyle\leq$
$\displaystyle f\left(\int_{\partial K}\frac{H_{n-1}}{\langle
x,N(x)\rangle^{n+1}}\langle
x,N(x)\rangle\frac{d\mu_{\vec{i}}}{\mu_{\vec{i}}-{\rm
vol}_{n}(K)}\right)\,\mu_{\vec{i}}-{\rm vol}_{n}(K)$ $\displaystyle=$
$\displaystyle f\left(\frac{\mu_{\vec{i}}-{\rm
vol}_{n}(K^{\circ}}{\mu_{\vec{i}}-{\rm vol}_{n}(K)}\right)\,\mu_{\vec{i}}-{\rm
vol}_{n}(K).$

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

$\displaystyle D_{f}(P_{K},Q_{K})$ $\displaystyle=$
$\displaystyle\int_{\partial
K}\left(a\frac{p_{K}}{q_{K}}+b\right)q_{K}\,d\mu_{\vec{i}}=a\int_{\partial
K}p_{K}\,d\mu_{\vec{i}}+b\int_{\partial K}q_{K}\,d\mu_{\vec{i}}$
$\displaystyle=$ $\displaystyle a\,\mu_{\vec{i}}-{\rm
vol}_{n}(K^{\circ})+b\,\mu_{\vec{i}}-{\rm vol}_{n}(K).$

If $f$ is not linear, equality holds iff $\frac{p_{K}}{q_{K}}=c$, $c$ 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 that $K\cup L$ is convex

$D_{f}({P}_{K\cup L},{Q}_{K\cup L})+D_{f}({P}_{K\cap L},{Q}_{K\cap
L})=D_{f}({P}_{K},{Q}_{K})+D_{f}({P}_{L},{Q}_{L}).$ (27)

###### Proposition 4.3

Let $K$ be a convex body in $\mathbb{R}^{n}$ with the origin in its interior
and let $f:(0,\infty)\rightarrow\mathbb{R}$ be a convex function. Then
$D_{f}(P_{K},Q_{K})$ are valuations.

Proof. To prove (27), we proceed as in Schütt [29]. For completeness, we
include the argument. We decompose

$\partial(K\cup L)=(\partial K\cap\partial L)\cup(\partial K\cap
L^{c})\cup(K^{c}\cap\partial L),$ $\partial(K\cap L)=(\partial K\cap\partial
L)\cup(\partial K\cap\text{int}L)\cup(\text{int}K\cap\partial L),$ $\partial
K=(\partial K\cap\partial L)\cup(\partial K\cap L^{c})\cup(\partial
K\cap\text{int}L),$ $\partial L=(\partial K\cap\partial L)\cup(\partial
K^{c}\cap\partial L)\cup(\text{int}K\cap\partial L),$

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,

$\langle x,N_{K}(x)\rangle=\langle x,N_{L}(x)\rangle=\langle x,N_{K\cap
L}(x)\rangle=\langle x,N_{K\cup L}(x)\rangle$ (28)

and

$\kappa_{K\cup L}(x)=\min\\{\kappa_{K}(x),\kappa_{L}(x)\\},\ \ \kappa_{K\cap
L}(x)=\max\\{\kappa_{K}(x),\kappa_{L}(x)\\}.$ (29)

To prove (27), we split the involved integral using the above decompositions
and (28) and (29).

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Kateryna Tatarko,  Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1. e-mail<EMAIL_ADDRESS>|  Elisabeth M. Werner,  Department of Mathematics, Case Western Reserve University, Cleveland, Ohio, USA, 44106. e-mail<EMAIL_ADDRESS>
---|---


$\displaystyle\partial_{\theta_{1}}\partial_{\theta_{2}}\partial_{\bar{\theta}_{3}}\partial_{\bar{\theta}_{4}}\langle\Psi_{p_{1}}\Psi_{p_{2}}\Psi_{p_{3}}\Psi_{p_{4}}\rangle|_{\theta_{i}=\bar{\theta}_{i}=0}$
$\displaystyle=\frac{P_{p_{i}}}{x_{12}x_{34}y_{12}y_{34}}\left({\mathcal{D}}^{(p_{12},p_{43})}_{x}-{\mathcal{D}}^{(-p_{12},-p_{43})}_{y}\right)P_{p_{i}}^{-1}\frac{xy}{x-y}\langle\Psi_{p_{1}}\Psi_{p_{2}}\Psi_{p_{3}}\Psi_{p_{4}}\rangle|_{\theta_{i}=\bar{\theta}_{i}=0}$
$\displaystyle={\mathcal{I}}^{-1}{\mathcal{C}}_{1,2}^{SU(1,1|2)}\langle\Psi_{p_{1}}\Psi_{p_{2}}\Psi_{p_{3}}\Psi_{p_{4}}\rangle|_{\theta_{i}=\bar{\theta}_{i}=0}\
,$ (203)

with $\mathcal{I}$ defined in (49). This is precisely the result claimed in
(48). It would be interesting to understand in a more direct way precisely why
the Casimir appears in this context.

##  Appendix C Further Unmixing Examples

In the end of section 6.2 we described some simple examples of unmixing in the
2-derivative sector. In this Appendix, we will describe more complicated
examples. First we will describe unmixing in the 2-derivative sector with non-
zero $SU(2)$ charge $p$ and then we will describe some examples in the
6-derivative sector.

###  C.1 2-derivative sector

The simplest case with $p>0$ is $\left(\Delta,p\right)=(2,1)$, for which there
is only one exchanged operator in the double-trace spectrum,
${\mathcal{O}}_{1}{\mathcal{O}}_{2}$. Performing the conformal block expansion
of $H_{1212}$ and comparing to (6.2), we find that the coefficients for
$\left(\Delta,p\right)=(2,1)$ are

$\displaystyle A^{(0)}_{1212}(2,1)$ $\displaystyle=\frac{1}{4}\ ,\quad
M^{\text{(2)}}_{1212}(2,1)=1.$ (204)

The unmixing equations in (6.2) are then solved by

$\gamma^{\text{(2)}}_{2,1}=4\ ,\quad C^{(0)}_{2,1}=\frac{1}{2}\ .$ (205)

Next, let us solve the mixing problem for $(\Delta,p)=(4,1)$ where two types
of operators are exchanged, ${\mathcal{O}}_{1}\partial{\mathcal{O}}_{2}$ and
${\mathcal{O}}_{2}{\mathcal{O}}_{3}$. Performing the conformal block expansion
of the relevant correlators then gives

$\displaystyle\hat{A}^{(0)}_{4,1}=\begin{pmatrix}A^{(0)}_{1212}&0\\\
0&A^{(0)}_{2323}\end{pmatrix}_{(4,1)}=\begin{pmatrix}\frac{5}{84}&0\\\
0&\frac{1}{30}\end{pmatrix}\ ,\
\hat{M}^{\text{(2)}}_{4,1}=\begin{pmatrix}M^{\text{(2)}}_{1212}&M^{\text{(2)}}_{1223}\\\
M^{\text{(2)}}_{1223}&M^{\text{(2)}}_{2323}\end{pmatrix}_{(4,1)}=\begin{pmatrix}\frac{5}{21}&\frac{1}{3}\\\
\frac{1}{3}&\frac{7}{15}\end{pmatrix}\ .$ (206)

Solving the unmixing equations gives the following anomalous dimensions and
OPE coefficients:

$\displaystyle\gamma^{\text{(2)}}_{4,1}=18\
,\quad\mathbb{C}^{(0)}_{4,1}=\begin{pmatrix}\frac{\sqrt{5}}{3\sqrt{42}}&-\frac{\sqrt{5}}{6\sqrt{3}}\\\
\frac{\sqrt{7}}{3\sqrt{30}}&\frac{1}{3\sqrt{15}}\end{pmatrix}\ .$ (207)

Note that there is only one non-zero anomalous dimension, as expected.

###  C.2 6-derivative sector

Let us first consider $t=\Delta-p$ odd. For the simplest case
$\left(\Delta,p\right)=(1,0)$ only one exchanged operator
${\mathcal{O}}_{1}{\mathcal{O}}_{1}$ contributes. Performing the conformal
block expansion of $H^{\text{(6)}}_{1111}$ in (5.2), we find that the
coefficient for $\left(\Delta,p\right)=(1,0)$ is

$\displaystyle M^{\text{(6)}}_{1111}(1,0)=4\ .$ (208)

The unmixing equations are

$M^{\text{(6)}}_{1111}(1,0)=\gamma^{\text{(6)}}_{1,0}\left(C^{(0)}_{1,0}\right)^{2}+2\,\gamma^{\text{(2)}}_{1,0}C^{(0)}_{1,0}C^{\text{(6)}}_{1,0}\
,\quad 0=C^{(0)}_{1,0}C^{\text{(6)}}_{1,0}\\\ .$ (209)

We can solve these equations using (136), which yields

$\left(\gamma^{\text{(6)}}_{A}\right)^{1,0}_{0,0}=4\ ,\quad
C^{\text{(6)}}_{1,0}=0\ .$ (210)

For $\left(\Delta,p\right)=(3,0)$ there are two possible exchanged operators,
${\mathcal{O}}_{2}{\mathcal{O}}_{2}$ and
${\mathcal{O}}_{1}\partial^{2}{\mathcal{O}}_{1}$. In this case we find

$\displaystyle\hat{M}^{\text{(6)}}_{3,0}=\begin{pmatrix}M^{\text{4-deriv}}_{1111}&M^{\text{4-deriv}}_{1122}\\\
M^{\text{4-deriv}}_{1122}&M^{\text{4-deriv}}_{2222}\end{pmatrix}_{(3,0)}=\begin{pmatrix}\frac{72}{5}&\frac{4}{3}(15+\,C_{0})\\\
\frac{4}{3}(15+\,C_{0})&\frac{8}{9}(38+5\,C_{0})\end{pmatrix}.\ $ (211)

Solving the unmixing equations using (138) then gives

$\displaystyle\gamma^{\text{(6)}}_{3,0,i}$
$\displaystyle=\left\\{\frac{16}{3}(137+15C_{0}),\ \frac{64}{3}\right\\}\ ,\
\mathbb{C}^{\text{(6)}}_{3,0}=(-1+3C_{0})\begin{pmatrix}\frac{-\sqrt{5}}{9\sqrt{3}}&\frac{1}{9\sqrt{3}}\\\
\frac{\sqrt{5}}{27\sqrt{3}}&\frac{5}{27\sqrt{3}}\end{pmatrix}\ .$ (212)

Recall that $C_{0}$ is the coefficient of the ambiguity in (111). While there
is only one non-zero anomalous dimension for each $\left(\Delta,p\right)$ in
the 2-derivative sector, we see that there are two non-zero anomalous
dimensions in the 6-derivative sector for $(\Delta,p)=(3,0)$.

The first example where square roots appear in the 6-derivative sector is
$(\Delta,p)=(5,2)$. In this case there are four class $A$ operators,
$\left\\{{\mathcal{O}}_{1}\partial^{2}{\mathcal{O}}_{3},{\mathcal{O}}_{2}{\mathcal{O}}_{4},{\mathcal{O}}_{2}\partial^{2}{\mathcal{O}}_{2},{\mathcal{O}}_{3}{\mathcal{O}}_{3}\right\\}$,
and one class $B$ operator, ${\mathcal{O}}_{2}\partial{\mathcal{O}}_{3}$.
Class $A$ and $B$ operators do not mix with each other and the class $A$
operators will have square roots in their anomalous dimensions after unmixing.
Performing the conformal block expansion of the relevant correlators at
${\mathcal{O}}(a/c)$ then leads to the symmetric matrix:

$\displaystyle\hat{M}^{\text{(6)}}_{5,2}=\begin{pmatrix}M^{\text{(6)}}_{1313}&M^{\text{(6)}}_{1324}&M^{\text{(6)}}_{1322}&M^{\text{(6)}}_{1333}&M^{\text{(6)}}_{1323}\\\
&M^{\text{(6)}}_{2424}&M^{\text{(6)}}_{2422}&M^{\text{(6)}}_{2433}&M^{\text{(6)}}_{2423}\\\
&&M^{\text{(6)}}_{2222}&M^{\text{(6)}}_{2233}&M^{\text{(6)}}_{2223}\\\
&&&M^{\text{(6)}}_{3333}&M^{\text{(6)}}_{3323}\\\
&&&&M^{\text{(6)}}_{2323}\end{pmatrix}_{(5,2)}$
$\displaystyle=\begin{pmatrix}\frac{2}{25}(774+35C_{0})&\frac{6}{35}(416+35C_{0})&\frac{4}{45}(942+35C_{0})&\frac{36}{35}(90+7C_{0})&0\\\
&\frac{6}{245}(3910+441C_{0})&\frac{36}{35}(90+7C_{0})&\frac{468}{245}(62+7C_{0})&0\\\
&&\frac{8}{135}(1955+56C_{0})&\frac{8}{105}(1707+112C_{0})&0\\\
&&&\frac{72}{245}(657+56C_{0})&0\\\ &&&&\frac{672}{25}\end{pmatrix}\ .$ (213)

Solving the unmixing equations then gives

$\displaystyle\left(\gamma^{\text{(6)}}_{A}\right)^{5,2}_{0,0}=$
$\displaystyle\frac{32}{3}(545+51C_{0})\
,\quad\left(\gamma^{\text{(6)}}_{A}\right)^{5,2}_{1,0}=\frac{8}{15}(415-\sqrt{2881})\
,$ $\displaystyle\left(\gamma^{\text{(6)}}_{A}\right)^{5,2}_{0,1}=$
$\displaystyle\frac{8}{15}(415+\sqrt{2881})\
,\quad\left(\gamma^{\text{(6)}}_{B^{\text{to}}}\right)^{5,2}_{1}=\frac{1008}{5}\
.$ (214)

Note the appearance of square roots, which signal the breaking of 4d conformal
symmetry.

Finally, let us describe a few examples of unmixing for $t=\Delta-p$ even. At
weight $(\Delta,p)=(3,1)$ there is only one operator present,
${\mathcal{O}}_{1}\partial{\mathcal{O}}_{2}$, and we get

$\displaystyle A^{(0)}_{1212}(3,1)$ $\displaystyle=\frac{2}{15}\ ,\
M^{\text{(6)}}_{1212}(3,1)=\frac{16}{9}\ .$ (215)

Solving the unmixing equations then gives the anomalous dimension and OPE
coefficients

$\left(\gamma^{\text{(6)}}_{B^{\text{te}}}\right)^{3,1}_{1,0}=\frac{40}{3}\
,\quad C^{(0)}_{3,1}=\sqrt{\frac{2}{15}}\ ,\quad C^{\text{(6)}}_{3,1}=0\ .$
(216)

For $(\Delta,p)=(5,1)$ there are two operators that contribute,
${\mathcal{O}}_{1}\partial^{2}{\mathcal{O}}_{2}$ and
${\mathcal{O}}_{2}\partial{\mathcal{O}}_{3}$, and performing the conformal
block expansion of the relevant correlators gives

$\displaystyle\hat{A}^{(0)}_{5,1}$
$\displaystyle=\begin{pmatrix}A^{(0)}_{1212}&0\\\ 0&A^{(0)}_{2323}\\\
\end{pmatrix}_{(5,1)}=\begin{pmatrix}\frac{1}{42}&0\\\
0&\frac{9}{175}\end{pmatrix}\ ,$ $\displaystyle\hat{M}^{\text{(6)}}_{5,1}$
$\displaystyle=\begin{pmatrix}M^{\text{(6)}}_{1212}&M^{\text{(6)}}_{1223}\\\
M^{\text{(6)}}_{1223}&M^{\text{(6)}}_{2323}\\\
\end{pmatrix}_{(5,1)}=\begin{pmatrix}\frac{52}{45}&\frac{104}{25}\\\
\frac{104}{25}&\frac{1872}{125}\end{pmatrix}\ .$ (217)

Solving the unmixing equations for this case gives the anomalous dimension

$\left(\gamma^{\text{(6)}}_{B^{\text{te}}}\right)^{5,1}_{1,0}=\frac{5096}{15}\
.$ (218)

For this case again, as for the whole $p=1$ sector, there is exactly one non-
zero anomalous dimension, which corresponds to the operator with
$i_{B^{\text{te}}}=1,\ r_{B^{\text{te}}}=0$. Going to $p\geq 2$, there will be
two non-zero anomalous dimensions, the additional one being for the operator
with $i_{B^{\text{te}}}=1,\ r_{B^{\text{te}}}=1$.

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