polarized toric manifold $(X_{P},L_{P})$ satisfying (4.5) for any
$f\in\mathcal{C}_{PL}^{\mathbf{Q}}$ is uniformly relatively K-polystable.

##  5\. Proof of Theorem 1.0.1

###  5.1. Spaces of convex functions

Let $P\subset M_{\mathbf{R}}$ be an $n$-dimensional integral Delzant polytope
defined by

$\displaystyle P=\\{x\in
M_{\mathbf{R}}\mid\langle\lambda_{j},x\rangle+d_{j}\geq 0\ (j=1,\ldots,r)\\},$

where $\lambda_{j}\in N$, $d_{j}\in\mathbf{Z}$, $r$ is the number of the
facets, and each $\lambda_{j}$ is primitive. Throughout this section, we
assume the origin 0 lies in the interior of $P$. Then $d_{j}>0$ for any
$j=1,\ldots,r$. Let $\mathrm{CVX}(P)$ denote the set of all lower
semicontinuous convex functions $f\colon P\to(-\infty,+\infty]$ such that
$f\not\equiv+\infty$. Note that every function in $\mathrm{CVX}(P)$ is bounded
from below since $P$ is compact. For each subset $\mathcal{F}$ of
$\mathrm{CVX}(P)$, set

$\displaystyle\widetilde{\mathcal{F}}\coloneqq\\{f\in\mathcal{F}\mid\inf_{P}f=f(0)=0\\}.$

If $\mathcal{F}$ is closed under addition of affine functions, then for any
function $f\in\mathcal{F}$ there exists an affine function $\ell$ such that
$f+\ell\in\widetilde{\mathcal{F}}$ by the supporting hyperplane theorem.

Let $\mathcal{E}_{1}\coloneqq\mathrm{CVX}(P)\cap L^{1}(P)$. By the toric
pluripotential theory [24, Proposition 3.2, Theorem 3.6, Proposition 3.9],
$\mathcal{E}_{1}$ can be identified with the metric completion of the space
$(\mathcal{S},d_{1})$. For each $j=1,\ldots,r$, let $F_{j}$ denote the facet
of $P$ defined by

$\displaystyle F_{j}=\\{x\in P\mid\langle\lambda_{j},x\rangle+d_{j}=0\\},$

and define a subset $P^{\ast}$ of $P$ by

$\displaystyle
P^{\ast}=P^{\circ}\cup\left(\bigcup_{j=1}^{r}F_{j}^{\circ}\right).$

Here $F_{j}^{\circ}$ denotes the relative interior of $F_{j}$. Let
$\mathcal{C}_{\ast}$ be the set of all elements of $\mathrm{CVX}(P)$ which are
continuous on $P^{\ast}$ and integrable on $\partial P$. Note that the
integral of any function in $\mathcal{C}_{\ast}$ over the boundary makes sense
since the measure $\sigma$ is supported on the facets of $P$. A relation
between $\mathcal{C}_{\ast}$ and $\mathcal{E}_{1}$ is given by the following
proposition.

######  Proposition 5.1.1.

$\mathcal{C}_{\ast}=\mathcal{E}_{1}\cap L^{1}(\partial P)$.

We explain a proof of Proposition 5.1.1 below. The inclusion
$\mathcal{C}_{\ast}\subset\mathcal{E}_{1}\cap L^{1}(\partial P)$ is obtained
from the following proposition.

######  Proposition 5.1.2.

Let $d=\max\\{d_{1},\ldots,d_{r}\\}$. If $f$ is a nonnegative function in
$\mathcal{C}_{\ast}$, then

$\displaystyle\int_{P}f\,dx\leq\frac{d}{n+1}\left(\frac{\sigma(\partial
P)}{n}f(0)+\int_{\partial P}f\,d\sigma\right).$

In particular, for any $f\in\widetilde{\mathcal{C}}_{\ast}$ we have

$\displaystyle\int_{P}f\,dx\leq\frac{d}{n+1}\int_{\partial P}f\,d\sigma.$

###### Proof.

For each $j=1,\ldots,r$, let $C(F_{j})=\\{t\zeta\mid t\in[0,1],\ \zeta\in
F_{j}\\}$. Then we have

$\displaystyle P=\bigcup_{j=1}^{r}C(F_{j}).$

Also, by convexity of $f$ we have

$\displaystyle\int_{C(F_{j})}f\,dx$ $\displaystyle\leq
d_{j}\int_{0}^{1}t^{n-1}\,dt\int_{F_{j}}((1-t)f(0)+tf)\,d\sigma$
$\displaystyle=d_{j}\int_{0}^{1}(1-t)t^{n-1}\,dt\int_{F_{j}}f(0)\,d\sigma+d_{j}\int_{0}^{1}t^{n}\,dt\int_{F_{j}}f\,d\sigma$
$\displaystyle=\frac{d_{j}}{n+1}\left(\frac{\sigma(F_{j})}{n}f(0)+\int_{F_{j}}f\,d\sigma\right).$

Hence we obtain

$\displaystyle\int_{P}f\,dx$
$\displaystyle=\sum_{j=1}^{r}\int_{C(F_{j})}f\,dx$
$\displaystyle\leq\sum_{j=1}^{r}\frac{d_{j}}{n+1}\left(\frac{\sigma(F_{j})}{n}f(0)+\int_{F_{j}}f\,d\sigma\right)$
$\displaystyle\leq\frac{d}{n+1}\left(\frac{\sigma(\partial
P)}{n}f(0)+\int_{\partial P}f\,d\sigma\right).$

The latter claim is now obvious. ∎

######  Corollary 5.1.3.

Every function in $\mathcal{C}_{\ast}$ is integrable on $P$. In particular, we
have $\mathcal{C}_{\ast}\subset\mathcal{E}_{1}\cap L^{1}(\partial P)$.

###### Proof.

Let $f\in\mathcal{C}_{\ast}$. Then there is an affine function $\ell$ so that
$\tilde{f}\coloneqq f+\ell\in\widetilde{\mathcal{C}}_{*}$. By Proposition
5.1.2, we have

$\displaystyle\int_{P}\tilde{f}\,dx\leq\frac{d}{n+1}\int_{\partial
P}\tilde{f}\,d\sigma<\infty$

and hence $\tilde{f}$ is integrable on $P$. Since $\ell$ is integrable on $P$,
$f=\tilde{f}-\ell$ is also integrable on $P$. ∎

Let us show the converse inclusion. For the purpose, we use the following
facts from the Donldson’s work [34]. Let $\Omega$ be a bounded convex open set
of $\mathbf{R}^{n}$, and $f\colon\Omega\to\mathbf{R}$ be a convex function.
For each $x\in\Omega$, set

$\displaystyle D_{x}(f)\coloneqq\sup\\{|\lambda|\mid
f(y)\geq\langle\lambda,x\rangle+f(y)\ (y\in\Omega)\\}.$

######  Proposition 5.1.4 ([34, Lemma 5.2.3]).

There is a constant $\kappa>0$ such that if a nonnegative convex function $f$
on $\Omega$ is integrable, then

$\displaystyle D_{x}(f)\leq\kappa d_{x}^{-(n+1)}\int_{\Omega}f\,dy$

for any $x\in\Omega$. Here $d_{x}$ is the distance from $x$ to the boundary of
$\Omega$.

######  Proposition 5.1.5 ([34, Lemma 5.2.4]).

For any convex function $f\colon\Omega\to\mathbf{R}$ and any two points
$x,y\in P$,

$\displaystyle|f(x)-f(y)|\leq\max\\{D_{x}(f),D_{y}(f)\\}|x-y|.$

Let $f\in\mathcal{E}_{1}\cap L^{1}(\partial P)$. By adding a suitable affine
function, we may assume $f\in\widetilde{\mathcal{E}_{1}}$. Since $f$ is
integrable over $P$, $f<+\infty$ on $P^{\circ}$. Continuity of $f$ on
$P^{\circ}$ is obvious by the convexity. Let $F\in\\{F_{1},\ldots,F_{r}\\}$.

######  Lemma 5.1.6.

For each $\zeta\in F$ we have ${\displaystyle f(\zeta)=\lim_{t\to
1}f(t\zeta)}$.

###### Proof.

Let $\zeta\in F$ and $s,t\in[0,1]$. If $s<t$, then we have

$\displaystyle\frac{f(t\zeta)-f(s\zeta)}{t-s}\geq\frac{f(t\zeta)-f(0)}{t-0}\geq
0$

by the convexity of $f$. Hence the function $[0,1]\ni t\mapsto
f(t\zeta)\in\mathbf{R}$ is monotonically non-decreasing. Further, since

$\displaystyle f(t\zeta)\leq(1-t)f(0)+tf(\zeta)=tf(\zeta)$

for any $t\in[0,1]$, we obtain

$\displaystyle\lim_{t\to 1}f(t\zeta)\leq
f(\zeta)\leq\liminf_{\xi\to\zeta}f(\xi)$

by the lower semicontinuity of $f$. We claim that
$\liminf_{\xi\to\zeta}f(\xi)\leq\lim_{t\to 1}f(t\zeta)$. Let $\delta>0$ and
$t\in[0,1]$, and suppose $|t-1|<\delta/|\zeta|$. Then $t\zeta\in
B(\zeta,\delta)$ and

$\displaystyle\inf_{B(\zeta,\delta)\cap P}f\leq f(t\zeta).$

Hence we have

$\displaystyle\inf_{B(\zeta,\delta)\cap P}f\leq\lim_{t\to 1}f(t\zeta)$

and

$\displaystyle\liminf_{\xi\to\zeta}f(\xi)=\sup_{\delta>0}\inf_{B(\zeta,\delta)\cap
P}f\leq\lim_{t\to 1}f(t\zeta),$

as required. ∎

For each $\eta\in(0,1]$, define a Borel measure $\sigma_{\eta}$ on
$\eta(\partial P)$ by

$\displaystyle\sigma_{\eta}(A)\coloneqq\eta^{n-1}\sigma(\eta^{-1}A).$

By setting $C\coloneqq\int_{\partial P}f(\xi)\,d\sigma$, we have

$\displaystyle\int_{\eta(\partial
P)}f(\xi)\,d\sigma_{\eta}=\eta^{n-1}\int_{\partial
P}f(\eta\xi)\,d\sigma\leq\eta^{n}\int_{\partial P}f(\xi)\,d\sigma=\eta^{n}C.$

Let $K$ be a compact subset of $F^{\circ}$. We write $F$ as

$\displaystyle F=\\{x\in P\mid\langle\lambda,x\rangle+d=0\\}.$

Regarding $f$ as a nonnegative convex function on $\eta F^{\circ}$, by
Proposition 5.1.4 there is a constant $\kappa>0$ such that

$\displaystyle D_{\zeta}(f)\leq|\lambda|\kappa d_{\zeta}^{-n}\int_{\eta
F}f(\xi)\,d\sigma_{\eta}\leq|\lambda|\kappa d_{\zeta}^{-n}\eta^{n}C$

for any $\zeta\in\eta K$. Here $d_{\zeta}$ is the distance from $\zeta$ to the
boundary $\partial(\eta F^{\circ})$. Also, by setting $d_{\eta K}$ with the
distance between $\eta K$ and $\partial(\eta F^{\circ})$, we have

$\displaystyle D_{\zeta}(f)\leq|\lambda|\kappa
d_{\zeta}^{-n}\eta^{n}C\leq|\lambda|\kappa d_{\eta
K}^{-n}\eta^{n}C=|\lambda|\kappa d_{K}^{-n}C.$

Hence, by Proposition 5.1.5 we obtain

$\displaystyle|f(\zeta^{\prime})-f(\zeta^{\prime\prime})|\leq\max\\{D_{\zeta^{\prime}}(f),D_{\zeta^{\prime\prime}}(f)\\}|\zeta^{\prime}-\zeta^{\prime\prime}|\leq|\lambda|\kappa
d_{K}^{-n}C|\zeta^{\prime}-\zeta^{\prime\prime}|$

for any $\zeta^{\prime},\zeta^{\prime\prime}\in\eta K$. Let $\zeta\in
F^{\circ}$, and

$\displaystyle K\coloneqq\overline{B(\zeta,\delta|\zeta|)}\cap\partial P$

for each $\delta>0$. By choosing $\delta$ to be sufficiently small, we may
assume that $K\subset F^{\circ}$. Set $C(K)\coloneqq\\{t\xi\mid t\geq 0,\xi\in
K\\}$. Then $C(K)$ is a closed neighborhood of $\zeta$ in $M_{\mathbf{R}}$.
Let $x\in C(K)\cap P^{\circ}$, and $H_{x}$ be the hyperplane of
$M_{\mathbf{R}}$ which contains $x$ and is parallel to the hyperplane
containing $F$. Then $H_{x}$ intersects the line $\mathbf{R}\zeta$ at a single
point $\\{y\\}$. Define $t(x)\in[0,1]$ by $y=t(x)\zeta$. Then the function
$x\mapsto t(x)$ is continuous and $t(x)\to 1$ (as $x\to\zeta$). Also, since
$x,y\in t(x)K$ we have

$\displaystyle|f(x)-f(y)|\leq|\lambda|\kappa d_{K}^{-n}C|x-y|.$

Hence we obtain

$\displaystyle|f(\zeta)-f(x)|$ $\displaystyle\leq|f(\zeta)-f(y)|+|f(y)-f(x)|$
$\displaystyle\leq|f(\zeta)-f(t(x)\zeta)|+|\lambda|\kappa d_{K}^{-n}C|y-x|$
$\displaystyle\leq|f(\zeta)-f(t(x)\zeta)|+|\lambda|\kappa
d_{K}^{-n}C|t(x)\zeta-x|$ $\displaystyle\to 0\quad\text{(as $x\to\zeta$)}$

by Lemma 5.1.6. This shows that $f$ is continuous at $\zeta$, and completes
the proof of Proposition 5.1.1.

###  5.2. Approximations and compactness of convex functions

We collect various approximation and compactness results for convex functions,
which is crucial for our proof of Theorem 1.0.1.

######  Proposition 5.2.1.

Let $\\{f_{i}\\}_{i=1}^{\infty}$ be a sequence of nonnegative functions in
$\mathcal{C}_{*}$, and $f\in\mathcal{C}_{*}$. Suppose that
$\\{f_{i}\\}_{i=1}^{\infty}$ and $f$ satisfy the following conditions:

  1. (i)

$\\{f_{i}\\}_{i=1}^{\infty}$ converges locally uniformly to $f$ on
$P^{\circ}$.

  2. (ii)

${\displaystyle\sup_{i\in\mathbf{Z}_{>0}}\int_{\partial
P}f_{i}\,d\sigma<\infty}$.

Then ${\displaystyle\lim_{i\to\infty}\|f-f_{i}\|_{L^{1}(P)}=0}$.

###### Proof.

By assumption, there is a constant $C>0$ such that

(5.1)
$\displaystyle\begin{split}f(0)+\sup_{i\in\mathbf{Z}_{>0}}f_{i}(0)+\int_{\partial
P}f\,d\sigma+\sup_{i\in\mathbf{Z}_{>0}}\int_{\partial P}f_{i}\,d\sigma\leq
C.\end{split}$

Let $\eta\in(0,1)$. Then for each $i\in\mathbf{Z}_{>0}$ we have

$\displaystyle\|f-f_{i}\|_{L^{1}(P)}=\int_{P\setminus\eta
P}|f-f_{i}|\,dx+\int_{\eta P}|f-f_{i}|\,dx.$

By the convexity of $f_{i}$ and $f$, we have

$\displaystyle\int_{P\setminus\eta P}|f-f_{i}|\,dx$
$\displaystyle\leq\int_{P\setminus\eta P}(f+f_{i})\,dx$
$\displaystyle\leq\sum_{j=1}^{r}d_{j}\int_{\eta}^{1}(1-t)t^{n-1}\,dt\int_{F_{j}}(f(0)+f_{i}(0))\,d\sigma$
$\displaystyle\quad+\sum_{j=1}^{r}d_{j}\int_{\eta}^{1}t^{n}\,dt\int_{F_{j}}(f+f_{i})\,d\sigma$
$\displaystyle\leq 2Cd\sigma(\partial
P)\int_{\eta}^{1}(1-t)t^{n-1}\,dt+2Cd\int_{\eta}^{1}t^{n}\,dt$
$\displaystyle\leq 2Cd(\sigma(\partial P)+1)\int_{\eta}^{1}t^{n-1}\,dt$
$\displaystyle=2Cd(\sigma(\partial P)+1)\frac{1-\eta^{n}}{n}.$

For each $\varepsilon>0$, choose $\eta\in(0,1)$ so that

$\displaystyle 2Cd(\sigma(\partial
P)+1)\frac{1-\eta^{n}}{n}<\frac{\varepsilon}{2}.$

Since $\\{f_{i}\\}_{i=1}^{\infty}$ converges uniformly to $f$ on $\eta P$,
there exists $N\in\mathbf{Z}_{>0}$ such that for any integer $i\geq N$

$\displaystyle\int_{\eta P}|f-f_{i}|\,dx<\frac{\varepsilon}{2}.$

Hence, for any integer $i\geq N$ we have

$\displaystyle\|f-f_{i}\|_{L^{1}(P)}$ $\displaystyle=\int_{P\setminus\eta
P}|f-f_{i}|\,dx+\int_{\eta P}|f-f_{i}|\,dx<\varepsilon,$

as required. ∎

By using Proposition 5.2.1, we obtain the following improvements of [16, Lemma
3.1] and [34, Proposition 5.2.6].

######  Proposition 5.2.2 (cf. [16, Lemma 3.1]).

Let $f\in\widetilde{\mathcal{C}}_{\ast}$. Then there exists a sequence
$\\{f_{i}\\}_{i=1}^{\infty}$ in $\mathcal{C}_{\infty}\cap C^{\infty}(P)$ such
that

  1. (i)

$0\leq f_{i}$ for any $i\in\mathbf{Z}_{>0}$,

  2. (ii)

$\\{f_{i}\\}_{i=1}^{\infty}$ converges locally uniformly to $f$ on
$P^{\circ}$, and

  3. (iii)

${\displaystyle\int_{P}|f-f_{i}|\,dx+\int_{\partial P}|f-f_{i}|\,d\sigma\to
0}$ (as $i\to\infty$).

In particular, we have
${\displaystyle\lim_{i\to\infty}L_{V}(f_{i})=L_{V}(f)}$.

###### Proof.

By [16, Lemma 3.1], there exists a sequence $\\{f_{i}\\}_{i=1}^{\infty}$ in
$\mathcal{C}_{\infty}$ which satisfies (i), (ii), and

(5.2) $\displaystyle\lim_{i\to\infty}\int_{\partial P}|f-f_{i}|\,d\sigma=0.$

Moreover, a careful reading of the proof in [16] shows that we can choose
$\\{f_{i}\\}_{i=1}^{\infty}$ so that $f_{i}\in\mathcal{C}_{\infty}\cap
C^{\infty}(P)$ for any $i\in\mathbf{Z}_{>0}$. Then the sequence
$\\{f_{i}\\}_{i=1}^{\infty}$ satisfies the conditions (i), (ii) of Proposition
5.2.1, and hence it converges to $f$ in $L^{1}$. ∎

######  Proposition 5.2.3 (cf. [34, Proposition 5.2.6]).

Let $\\{f_{i}\\}_{i=1}^{\infty}$ be a sequence in
$\widetilde{\mathcal{C}}_{*}$ with

$\displaystyle\sup_{i\in\mathbf{Z}_{>0}}\int_{\partial
P}f_{i}\,d\sigma<\infty.$

Then there exists a subsequence $\\{f_{i_{k}}\\}_{k=1}^{\infty}$ of
$\\{f_{i}\\}_{i=1}^{\infty}$ and $f\in\widetilde{\mathcal{C}}_{*}$ such that

  1. (i)

$\\{f_{i_{k}}\\}_{k=1}^{\infty}$ converges locally uniformly to $f$ on
$P^{\circ}$, and

  2. (ii)

${\displaystyle\int_{P}|f-f_{i_{k}}|\,dx\to 0\ (\text{as $k\to\infty$})}$,
${\displaystyle\int_{\partial
P}f\,d\sigma\leq\liminf_{k\to\infty}\int_{\partial P}f_{i_{k}}\,d\sigma}$.

In particular, we have ${\displaystyle
L_{V}(f)\leq\liminf_{k\to\infty}L_{V}(f_{i_{k}})}$.

###### Proof.

By [34, Proposition 5.2.6], there exist a subsequence
$\\{f_{i_{k}}\\}_{k=1}^{\infty}$ of $\\{f_{i}\\}_{i=1}^{\infty}$ and
$f\in\widetilde{\mathcal{C}}_{*}$ which satisfy (i) and

$\displaystyle\int_{\partial
P}f\,d\sigma\leq\liminf_{k\to\infty}\int_{\partial P}f_{i_{k}}\,d\sigma.$

Then Proposition 5.2.1 shows that the sequence
$\\{f_{i_{k}}\\}_{k=1}^{\infty}$ converges to $f$ in $L^{1}$. The proposition
is proved. ∎

We also use the following approximation result due to Donaldson [34].

######  Proposition 5.2.4 ([34, Proposition 5.2.8]).

Let $f$ be a nonnegative function in $\mathcal{C}_{\ast}$. Then there exists a
sequence $\\{f_{i}\\}_{i=1}^{\infty}$ in $\mathcal{C}_{PL}$ such that

  1. (i)

$0\leq f_{i}\leq f$ for any $i\in\mathbf{Z}_{>0}$,

  2. (ii)

$\\{f_{i}\\}_{i=1}^{\infty}$ converges locally uniformly to $f$ on
$P^{\circ}$, and

  3. (iii)

${\displaystyle\int_{P}|f-f_{i}|\,dx+\int_{\partial P}|f-f_{i}|\,d\sigma\to
0}$ (as $i\to\infty$).

In particular, we have
${\displaystyle\lim_{i\to\infty}L_{V}(f_{i})=L_{V}(f)}$.

The following proposition is easily proved from the density of $\mathbf{Q}$ in
$\mathbf{R}$ and convexity.

######  Proposition 5.2.5.

Let $f\in\mathcal{C}_{PL}$ (possibly irrational). Then there exists a sequence
$\\{f_{i}\\}_{i=1}^{\infty}$ in $\mathcal{C}_{PL}^{\mathbf{Q}}$ which
converges uniformly to $f$ in $P$.

###### Proof.

Choose affine functions $\ell_{1},\ldots,\ell_{r}$ so that

$\displaystyle f(x)=\max\\{\ell_{1}(x),\ldots,\ell_{m}(x)\\}$

for any $x\in P$, and we regard $f$ as a function defined on $\mathbf{R}^{n}$.
For any $j\in\\{1,,\ldots,m\\}$, choose a sequence
$\\{\ell_{j}^{(i)}\\}_{i=1}^{\infty}$ of rational affine functions which
converges pointwise to $\ell_{j}$ on $\mathbf{R}^{n}$, and define a rational
piecewise affine function $f^{(i)}$ by

$\displaystyle
f^{(i)}(x)=\max\\{\ell_{1}^{(i)}(x),\ldots,\ell_{m}^{(i)}(x)\\}.$

Then $\\{f^{(i)}\\}_{i=1}^{\infty}$ converges locally uniformly to $f$ on
$\mathbf{R}^{n}$. In particular, $\\{f^{(i)}\\}_{i=1}^{\infty}$ converges
uniformly to $f$ on $P$. ∎

###  5.3. The non-Archimedean K-energy

We can define $L(f)$ and $L_{V}(f)$ for any $f\in\mathcal{E}_{1}$, taking
value in $(-\infty,\infty]$. By using these, we can get the following
characterization of $\mathcal{C}_{\ast}$.

######  Proposition 5.3.1.

$\displaystyle\mathcal{C}_{\ast}=\\{f\in\mathcal{E}_{1}\mid
L(f)<\infty\\}=\\{f\in\mathcal{E}_{1}\mid L_{V}(f)<\infty\\}.$

###### Proof.

By Proposition 5.1.1, for each $f\in\mathcal{E}_{1}$ we have

$\displaystyle f\in\mathcal{C}_{\ast}\iff\int_{\partial
P}f\,d\sigma<\infty\iff L(f)<\infty\iff L_{V}(f)<\infty,$

as required. ∎

###  5.4. The J-norm

For any convex function $f$ in $\mathcal{E}_{1}$, define

$\displaystyle\|f\|_{J}$
$\displaystyle=\inf_{\text{$\xi\colon$affine}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}(f+\xi)\,dx-\inf_{P}(f+\xi)\right).$

We call $\|f\|_{J}$ the _J-norm_ of $f$. The followings are the basic
properties on the J-norm.

######  Proposition 5.4.1.

  1. (1)

$\|f+\xi\|_{J}=\|f\|_{J}$ for any $f\in\mathcal{E}_{1}$ and $\xi\in
N_{\mathbf{R}}$.

  2. (2)

$\|f\|_{J}\geq 0$ for any $f\in\mathcal{E}_{1}$, and $\|f\|_{J}=0$ if and only
if $f$ is affine.

  3. (3)

There exists a constant $C_{1}>0$ such that

$\displaystyle\|f\|_{J}\leq\|f\|_{L^{1}(P)}\leq C_{1}\|f\|_{J}$

for any $f\in\widetilde{\mathcal{E}}_{1}$.

  4. (4)

If a sequence $\\{f_{i}\\}_{i=1}^{\infty}$ in $\mathcal{E}_{1}$ converges
uniformly to $f\in\mathcal{E}_{1}$, then

$\displaystyle\lim_{i\to\infty}\|f_{i}\|_{J}=\|f\|_{J}.$

###### Proof.

The claim (1) is clear, and (2) is easily deduced from (1) and (3). Let us
show the claim (3). The first inequality is obvious. For the latter
inequality, it is sufficient to show that there exists a constant $\kappa>0$
such that

$\displaystyle\|f\|_{J}\geq\kappa$

for any $f\in\widetilde{\mathcal{E}}_{1}$ with $\|f\|_{L^{1}(P)}=1$. For
contradiction, suppose that the conclusion does not hold. Then there is a
sequence $\\{f_{i}\\}_{i=1}^{\infty}$ in $\widetilde{\mathcal{E}}_{1}$ such
that

  1. (i)

$\|f_{i}\|_{L^{1}(P)}=1$ for any $i\in\mathbf{Z}_{>0}$, and

  2. (ii)

${\displaystyle\lim_{i\to\infty}\|f_{i}\|_{J}=0}$.

By condition (ii), there is a sequence of affine functions
$\\{\ell_{i}\\}_{i=1}^{\infty}$ such that

  1. (iii)

${\displaystyle\inf_{P}(f_{i}+\ell_{i})=0}$ for any $i\in\mathbf{Z}_{>0}$, and

  2. (iv)

${\displaystyle\int_{P}(f_{i}+\ell_{i})\,dx=0}$.

By conditions (i), (ii), (iii), and (iv) and [34, Corollary 5.2.5], we may
assume both $\\{f_{i}\\}_{i=1}^{\infty}$ and
$\\{f_{i}+\ell_{i}\\}_{i=1}^{\infty}$ converge locally uniformly on
$P^{\circ}$. Then the sequence $\\{\ell_{i}\\}_{i=1}^{\infty}$ converges
locally uniformly on $P^{\circ}$. Moreover, since every $\ell_{i}$ is affine,
the limit $\ell$ is also affine and the convergence $\ell_{i}\to\ell$ is
uniform. Let

$\displaystyle
f(x)\coloneqq\lim_{i\to\infty}f_{i}(x),\quad\ell(x)\coloneqq\lim_{i\to\infty}\ell_{i}(x)$

for each $x\in P^{\circ}$. By (iv) above, $f+\ell=0$ on $P^{\circ}$ and hence
$f$ is affine on $P^{\circ}$. Moreover, since $\inf_{P^{\circ}}f=f(0)=0$ we
obtain

$\displaystyle f=-\ell=0.$

Therefore, we have

$\displaystyle
1=\|f_{i}\|_{L^{1}(P)}\leq\|f_{i}+\ell_{i}\|_{L^{1}(P)}+\|-\ell_{i}\|_{L^{1}(P)}\to
0\quad(\text{as $i\to\infty$}),$

which is contradiction.

Finally, we show the claim (4). For each $\varepsilon>0$, choose
$N\in\mathbf{Z}_{>0}$ so that

$\displaystyle\sup_{P}|f-f_{i}|<\frac{\varepsilon}{4\mathrm{vol}(P)}$

for any $i\in\mathbf{Z}_{\geq N}$. Then for any $i\in\mathbf{Z}_{\geq N}$ and
an affine function $\ell$ we have

$\displaystyle\left|\left(\int_{P}(f+\ell)\,dx-\mathrm{vol}(P)\inf_{P}(f+\ell)\right)-\left(\int_{P}(f_{i}+\ell)\,dx-\mathrm{vol}(P)\inf_{P}(f_{i}+\ell)\right)\right|$
$\displaystyle\leq\left|\int_{P}(f+\ell)\,dx-\int_{P}(f+\ell)\,dx\right|+\mathrm{vol}(P)\left|\inf_{P}(f+\ell)-\inf_{P}(f_{i}+\ell)\Big{)}\right|$
$\displaystyle\leq\left|\int_{P}(f-f_{i})\,dx\right|+\mathrm{vol}(P)\sup_{P}|f-f_{i}|$
$\displaystyle<\frac{\varepsilon}{4}+\frac{\varepsilon}{4}=\frac{\varepsilon}{2}.$

By choosing $\ell$ so that

$\displaystyle\int_{P}(f+\ell)\,dx-\mathrm{vol}(P)\inf_{P}(f+\ell)-\|f\|_{J}<\frac{\varepsilon}{2},$

we obtain

(5.3)
$\displaystyle\begin{split}\|f_{i}\|_{J}&\leq\int_{P}(f_{i}+\ell)\,dx-\mathrm{vol}(P)\inf_{P}(f_{i}+\ell)\\\
&<\int_{P}(f+\ell)\,dx-\mathrm{vol}(P)\inf_{P}(f+\ell)+\frac{\varepsilon}{2}\\\
&<\|f\|_{J}+\varepsilon.\end{split}$

Similarly, we have

(5.4) $\displaystyle\|f\|_{J}<\|f_{i}\|_{J}+\varepsilon.$

By (5.3) and (5.4), we obtain $|\|f\|_{J}-\|f_{i}\|_{J}|<\varepsilon$, as
required. ∎

Similar to the J-norm, _the reduced $L^{1}$-norm_ of a convex function
$f\in\mathcal{E}_{1}$ is defined by

$\displaystyle\|f\|_{1,T}\coloneqq\inf_{\text{$\xi\colon$affine}}\int_{P}|(f+\xi)-(\overline{f+\xi})|\,dx.$

The similar claims as (1), (2), (3) of Proposition 5.4.1 are valid for the
$J$-norm. In particular, the reduced $L^{1}$-norm and the $J$-norm are
equivalent in the following sense.

######  Proposition 5.4.2.

There are constants $C_{2},C_{3}>0$ such that

$\displaystyle C_{2}\|f\|_{1,T}\leq\|f\|_{J}\leq C_{3}\|f\|_{1,T}$

for any $f\in\mathcal{E}_{1}$.

An advantage of the reduced $L^{1}$-norm is the following $L^{1}$-continuity.

######  Proposition 5.4.3.

If a sequence $\\{f_{i}\\}_{i=1}^{\infty}$ in $\mathcal{E}_{1}$ converges to
$f\in\mathcal{E}_{1}$ in $L^{1}$, then

$\displaystyle\lim_{i\to\infty}\|f_{i}\|_{1,T}=\|f\|_{1,T}.$

###### Proof.

For each $\varepsilon>0$, choose $N\in\mathbf{Z}_{>0}$ so that

$\displaystyle\|f-f_{i}\|_{L^{1}(P)}<\frac{\varepsilon}{2}$

for any $i\in\mathbf{Z}_{\geq N}$. Then for any $i\in\mathbf{Z}_{\geq N}$ and
an affine function $\ell$ we have

$\displaystyle|\|f+\ell\|_{L^{1}(P)}-\|f_{i}+\ell\|_{L^{1}(P)}|\leq\|f-f_{i}\|_{L^{1}(P)}<\frac{\varepsilon}{2}.$

By choosing $\ell$ so that

$\displaystyle|\|f+\ell\|_{L^{1}(P)}-\|f\|_{1,T}<\frac{\varepsilon}{2},$

we obtain

(5.5)
$\displaystyle\|f_{i}\|_{1,T}\leq\|f_{i}+\ell\|_{L^{1}(P)}<\|f+\ell\|_{L^{1}(P)}+\frac{\varepsilon}{2}<\|f\|_{1,T}+\varepsilon.$

Similarly, we have

(5.6) $\displaystyle\|f\|_{1,T}<\|f_{i}\|_{1,T}+\varepsilon.$

By (5.5) and (5.6), we obtain $|\|f\|_{T,1}-\|f_{i}\|_{T,1}|<\varepsilon$, as
required. ∎

It is not known whether the similar result as Proposition 5.4.3 are valid for
the $J$-norm or not.

###  5.5. Proof of Theorem 1.0.1

In this subsection, we give a proof of Theorem 1.0.1. For
$(b)_{\mathcal{F}}\Rightarrow(J)_{\mathcal{F}}$, let $\delta>0$ satisfying

$\displaystyle L_{V}(f)\geq\delta\int_{\partial P}f\,d\sigma$

for any $f\in\widetilde{\mathcal{F}}$. Let $f\in\mathcal{F}$, and choose an
affine function $\ell$ so that $f+\ell\in\widetilde{\mathcal{F}}$. Then, by
Propositions 5.1.2 and 5.4.1 we have

$\displaystyle L_{V}(f)$ $\displaystyle=L_{V}(f+\ell)\geq\delta\int_{\partial
P}(f+\ell)\,d\sigma$
$\displaystyle\geq\left(\frac{n+1}{d}\right)\delta\int_{P}(f+\ell)\,dx$
$\displaystyle\geq\left(\frac{n+1}{2d}\right)\delta\|f+\ell\|_{J}$
$\displaystyle=\left(\frac{n+1}{2d}\right)\delta\|f\|_{J},$

as required.

For $(\mathrm{K})_{\mathcal{C}_{\ast}}\Rightarrow(b)_{\mathcal{F}}$, first
note that

$\displaystyle L_{V}(f)\geq 0$

for any $f\in\widetilde{\mathcal{C}}_{*}$. For contradiction, suppose that $P$
does not satisfy $(b)_{\mathcal{F}}$. Then there exists a sequence
$\\{f_{i}\\}_{i=1}^{\infty}$ in $\widetilde{\mathcal{F}}$ such that

  1. (a)

${\displaystyle\int_{\partial P}f_{i}\,d\sigma=1}$ for any
$i\in\mathbf{Z}_{>0}$, and

  2. (b)

${\displaystyle\lim_{i\to\infty}L_{V}(f_{i})=0}$.

By the condition (a), we may further assume that there exists
$f\in\widetilde{\mathcal{C}}_{*}$ which satisfies the conditions (i), (ii) of
Proposition 5.2.3. Since

$\displaystyle 0\leq L_{V}(f)\leq\liminf_{i\to\infty}L_{V}(f_{i})=0,$

we have $L_{V}(f)=0$ and that $f$ is affine. Since
$f\in\widetilde{\mathcal{C}}_{*}$, $f$ must be $0$, which contradicts the
choice of $f$.

The equivalence
$(\mathrm{K})_{\mathcal{C}_{\ast}}\Leftrightarrow(\mathrm{K})_{\mathcal{E}_{1}}$
can be obtained as follows. The implication
$(\mathrm{K})_{\mathcal{E}_{1}}\Rightarrow(\mathrm{K})_{\mathcal{C}_{\ast}}$
is clear. Let us assume $(\mathrm{K})_{\mathcal{C}_{\ast}}$ and
$f\in\mathcal{E}_{1}$. If $f\in\mathcal{C}_{\ast}$, then $L_{V}(f)\geq 0$ and
$L_{V}(f)=0$ if and only if $f$ is affine by the assumption
$(\mathrm{K})_{\mathcal{C}_{\ast}}$. If
$f\in\mathcal{E}_{1}\setminus\mathcal{C}_{\ast}$, then $L_{V}(f)=+\infty>0$ by
Proposition 5.3.1.

Finally, let us show
$(J)_{\mathcal{F}}\Rightarrow(\mathrm{K})_{\mathcal{C}_{\ast}}$ in each case.

Case 1: For the case $\mathcal{F}=\mathcal{E}_{1}$ or
$\mathcal{F}=\mathcal{C}_{\ast}$, it is obvious from Propositions 5.3.1 and
5.4.1.

Case 2: Let $\mathcal{F}=\mathcal{C}_{PL}$. It is sufficient to show that
$L_{V}(f)\geq 0$ for any $f\in\widetilde{\mathcal{C}}_{*}$ and $L_{V}(f)=0$ if
and only if $f$ is affine. Choose a constant $\delta>0$ so that

$\displaystyle L_{V}(f)\geq\delta\|f\|_{J}$

for any $f\in\mathcal{C}_{PL}$. Then by Proposition 5.4.2 we have

$\displaystyle L_{V}(f)\geq\delta C_{2}\|f\|_{1,T}.$

Let $f\in\widetilde{\mathcal{C}}_{*}$, and choose a sequence
$\\{f_{i}\\}_{i=1}^{\infty}$ in $\mathcal{C}_{PL}$ satisfying the conditions
(i), (ii), (iii) of Proposition 5.2.4. By conditions (i) and (iii), we have
$f_{i}\in\widetilde{\mathcal{C}}_{PL}$ for any $i\in\mathbf{Z}_{>0}$ and
${\displaystyle\lim_{i\to\infty}L_{V}(f_{i})=L_{V}(f)}$. Since

$\displaystyle L_{V}(f_{i})\geq\delta C_{2}\|f_{i}\|_{1,T}\geq 0$

for any $i\in\mathbf{Z}_{>0}$, we have

$\displaystyle L_{V}(f)\geq\delta C_{2}\|f\|_{1,T}\geq 0$

by Proposition 5.4.3. Suppose $L_{V}(f)=0$. Then $\|f\|_{1,T}=0$ and hence $f$
is affine.

Case 3: Let $\mathcal{F}=\mathcal{C}_{PL}^{\mathbf{Q}}$. Then then we can
reduce to the Case 2 as follows. Choose a constant $\delta>0$ so that

$\displaystyle L_{V}(f)\geq\delta\|f\|_{J}$

for any $f\in\mathcal{C}_{PL}^{\mathbf{Q}}$. Let $f\in\mathcal{C}_{PL}$. By
Proposition 5.2.5, there is a sequence $\\{f_{i}\\}_{i=1}^{\infty}$ in
$\mathcal{C}_{PL}^{\mathbf{Q}}$ which converges uniformly to $f$ on $P$. Since

$\displaystyle L_{V}(f_{i})\geq\delta\|f_{i}\|_{J}$

for any $i\in\mathbf{Z}_{>0}$, we have

$\displaystyle L_{V}(f)\geq\delta\|f\|_{J}$

by Proposition 5.4.1. Hence $P$ satisfies the condition
$(J)_{\mathcal{C}_{PL}}$.

Case 4: Let $\mathcal{F}=\mathcal{C}_{\infty}$. Then, by using Proposition
5.2.2 in place of Proposition 5.2.4, we can also prove
$(J)_{\mathcal{C}_{\infty}}\Rightarrow(\mathrm{K})_{\mathcal{C}_{\ast}}$ by
the similar argument as Case 2.

Case 5: Finally, let $\mathcal{F}=\mathcal{S}$. As well as Case 2, it is
sufficient to show that $L_{V}(f)\geq 0$ for any
$f\in\widetilde{\mathcal{C}}_{*}$ and $L_{V}(f)=0$ if and only if $f$ is
affine. Choose $\delta>0$ so that

$\displaystyle L_{V}(u)\geq\delta\|u\|_{J}$

for any $u\in\mathcal{S}$. Then by Proposition 5.4.2 we have

$\displaystyle L_{V}(u)\geq\delta C_{2}\|u\|_{1,T}.$

Let $f\in\widetilde{\mathcal{C}}_{*}$, and choose a sequence
$\\{f_{i}\\}_{i=1}^{\infty}$ in $\mathcal{C}_{\infty}\cap C^{\infty}(P)$
satisfying the conditions (i), (ii), (iii) of Proposition 5.2.2. Fix
$u_{0}\in\widetilde{\mathcal{S}}$ arbitrarily. Then, for each
$i\in\mathbf{Z}_{>0}$ and $t\in(0,\infty)$, $u_{0}+tf_{i}\in\mathcal{S}$ and

$\displaystyle L_{V}\left(\frac{u_{0}+tf_{i}}{t}\right)\geq\delta
C_{2}\left\|\frac{u_{0}+tf_{i}}{t}\right\|_{1,T}\geq 0.$

By taking the limit as $t\to\infty$ and $i\to\infty$, we obtain

(5.7) $\displaystyle L_{V}(f)\geq\delta C_{2}\|f\|_{1,T}\geq 0.$

Now suppose $L_{V}(f)=0$. Then $\|f\|_{1,T}=0$ and hence $f$ is affine.

######  Remark 5.5.1.

Let $\mathcal{B}$ be the set of all bounded lower semicontinuous convex
functions defined on the whole of $P$. Then $\mathcal{B}$ is contained in
$\mathcal{C}_{\ast}$ and consequently $(\mathrm{K})_{\mathcal{C}_{\ast}}$
implies $(\mathrm{K})_{\mathcal{B}}$. The latter condtion
$(\mathrm{K})_{\mathcal{B}}$ can be found in [56].

##  6\. Proof of Theorem 1.0.5

Finally in this section, we prove Theorem 1.0.5. By the implication
$(b)_{\mathcal{C}_{PL}^{\mathbf{Q}}}\Rightarrow(J)_{\mathcal{C}_{PL}^{\mathbf{Q}}}$
in Theorem 1.0.1, it is enough to show that $P$ satisfies the condition
$(b)_{\mathcal{C}_{PL}^{\mathbf{Q}}}$. First suppose that $P$ satisfies

$\displaystyle\overline{s}+\max_{P}V\leq\frac{n+1}{d}.$

Define a constant $\delta\geq 0$ by

$\displaystyle\delta=1-\frac{d}{n+1}(\overline{s}+\max_{P}V).$

Then, by Proposition 5.1.2 we have

$\displaystyle L_{V}(f)$ $\displaystyle=\int_{\partial
P}f\,d\sigma-\int_{P}(\overline{s}+V)f\,dx$ $\displaystyle\geq\int_{\partial
P}f\,d\sigma-(\overline{s}+\max_{P}V)\int_{P}f\,dx$
$\displaystyle\geq\int_{\partial
P}f\,d\sigma-\frac{d}{n+1}(\overline{s}+\max_{P}V)\int_{\partial P}f\,d\sigma$
$\displaystyle=\delta\int_{\partial P}f\,d\sigma$

for any $f\in\widetilde{\mathcal{C}}_{PL}^{\mathbf{Q}}$. Hence, if $P$
satisfies (1.2) then $\delta>0$ and $P$ satisfies the condition
$(b)_{\mathcal{C}_{PL}^{\mathbf{Q}}}$. Now suppose that $P$ satisfies (1.3).
In this case $\delta$ might be $0$, but at least we have

$\displaystyle L_{V}(f)\geq\delta\int_{\partial P}f\,d\sigma\geq 0$

for any $f\in\widetilde{\mathcal{C}}_{PL}^{\mathbf{Q}}$. For contradiction,
suppose that our conclusion does not hold. Then there exists a sequence
$\\{f_{i}\\}_{i=1}^{\infty}$ in $\widetilde{\mathcal{C}}_{PL}^{\mathbf{Q}}$
such that

(6.1) $\displaystyle\int_{\partial P}f_{i}\,d\sigma=1$

and

(6.2) $\displaystyle\lim_{i\to\infty}L_{V}(f_{i})=0.$

Furthermore, we can find a subsequence $\\{f_{i_{k}}\\}_{k=1}^{\infty}$ of
$\\{f_{i}\\}_{i=1}^{\infty}$ and $f\in\widetilde{\mathcal{C}}_{*}$ satisfying
the conditions (i), (ii), (iii) of Proposition 5.2.3. Hence we have

(6.3) $\displaystyle
L_{V}(f)\leq\liminf_{k\to\infty}L_{V}(f_{i_{k}})=\lim_{k\to\infty}L_{V}(f_{i_{k}})=0.$

Now suppose $f\not\equiv 0$ on $P^{*}$. Then $U=\\{x\in P^{*}\mid f(x)>0\\}$
is a nonempty open set of $P^{*}$. Since $V$ is a nonconstant affine function,
it holds that $V<\max_{P}V$ almost everywhere on $P$. It follows that

$\displaystyle L_{V}(f)$ $\displaystyle=\int_{\partial
P}f\,d\sigma-\int_{P}(\overline{s}+V)f\,dx$ $\displaystyle>\int_{\partial
P}f\,d\sigma-(\overline{s}+\max_{P}V)\int_{U}f\,dx$
$\displaystyle\geq\delta\int_{\partial P}f\,d\sigma$ $\displaystyle\geq 0,$

which contradicts to (6.3). Hence we have $f\equiv 0$ on $P^{*}$, and

(6.4) $\displaystyle\lim_{k\to\infty}\int_{P}(\overline{s}+V)f_{i_{k}}\,dx=0.$

On the other hand, by (6.1) and (6.2) we have

$\displaystyle\lim_{k\to\infty}\int_{P}(\overline{s}+V)f_{i_{k}}\,dx=\lim_{k\to\infty}\left(\int_{\partial
P}f_{i_{k}}\,d\sigma-L_{V}(f_{i_{k}})\right)=1,$

which contradicts to (6.4).

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