# Correspondences between convex geometry and complex geometry

Brian Lehmann and Jian Xiao

###### Abstract

We present several analogies between convex geometry and the theory of
holomorphic line bundles on smooth projective varieties or Kähler manifolds.
We study the relation between positive products and mixed volumes. We define
and study a Blaschke addition for divisor classes and mixed divisor classes,
and prove new geometric inequalities for divisor classes. We also reinterpret
several classical convex geometry results in the context of algebraic
geometry: the Alexandrov body construction is the convex geometry version of
divisorial Zariski decomposition; Minkowski’s existence theorem is the convex
geometry version of the duality between the pseudo-effective cone of divisors
and the movable cone of curves.

  *     * Keywords. Alexandrov body; Convexity; Geometric inequalities; Mixed volume; Zariski decomposition; Positive product

2010 Mathematics Subject Classification. 14C20; 32Q15; 52A39

[Français]

Titre. Correspondances entre géométrie convexe et géométrie complexe Résumé.
Nous présentons plusieurs analogies entre la géométrie convexe et la théorie
des fibrés en droites sur les variétés projectives lisses ou kählériennes.
Nous étudions les relations entre produits positifs et volumes mixtes. Nous
définissons et étudions une somme de Blaschke pour les classes de diviseurs
(mixtes ou non), et nous établissons de nouvelles inégalités géométriques pour
les classes de diviseurs. Nous réinterprétons également plusieurs résultats
classiques de géométrie convexe en termes de géométrie algébrique : la
construction du corps d’Alexandrov est la version géométrie convexe de la
décomposition de Zariski divisorielle; de même, le théorème d’existence de
Minkowski est le pendant en géométrie convexe de la dualité entre cône des
diviseurs pseudo-effectifs et celui des courbes mobiles.

###### Contents

  1. 1\. Introduction
  2. 2\. Preliminaries
  3. 3\. Positive products and mixed volumes
  4. 4\. Alexandrov body and Zariski decompositions
  5. 5\. Morse type inequalities
  6. 6\. Cone dualities and Minkowski’s existence theorem
  7. 7\. Blaschke addition for (1,1)-classes and its applications
  8. 8\. Projection volumes and the Brascamp-Lieb inequality

##  1\. Introduction

Many deep results concerning holomorphic line bundles in complex geometry are
inspired by results in convex geometry. For example, the Khovanskii-Teissier
inequalities for $(1,1)$-classes are inspired by the Alexandrov-Fenchel
inequalities for convex bodies. Such results were first explored in [Tei82]
and [Kho89]; [Gro90] also contains many nice discussions on these relations.

In this paper we extend the dictionary between the geometry of line bundles in
complex geometry and the theory of convex bodies. First, we prove new results
about divisors motivated by results in convex geometry. These results
strengthen and clarify existing geometric inequalities in the literature. A
key role is played by the refined structure of the movable cone of curves we
obtained in our previous work [LX16b]. Second, we reinterpret several
classical convex geometry results in the context of algebraic geometry.
Although these convex geometry results are well understood, we feel that
complex geometers could benefit from this discussion.

The outline of this correspondence is as follows. On the complex geometry
side, let $X$ be a smooth projective variety (or compact Kähler manifold) of
dimension $n$. We have the vector space $H^{1,1}(X,\mathbb{R})$ whose integral
points parameterize Chern classes of holomorphic line bundles. Inside of this
space, we have cones $\operatorname{\overline{Eff}}^{1}(X)$,
$\operatorname{Mov}^{1}(X)$ which represent $(1,1)$-classes satisfying various
positivity conditions. On the convex geometry side, we consider

$V:=C^{0}(\mathbb{S}^{n-1})/L(\mathbb{S}^{n-1})$

where $\mathbb{S}^{n-1}$ is the $(n-1)$-dimensional sphere,
$C^{0}(\mathbb{S}^{n-1})$ is the space of continuous functions and
$L(\mathbb{S}^{n-1})$ is the space of linear functions. We consider two
positive cones in $V$: $\mathcal{C}_{p}$ is the cone generated by positive
continuous functions and $\mathcal{C}$ is the cone generated by positive
convex and homogeneous of degree one functions. We then study the following
correspondences:

| Complex Geometry | Convex Geometry
---|---|---
Section 3 |  | movable classes
---
and the positive products
$\mathcal{C}$ and the mixed volumes
Section 4 |  | pseudo-effective classes and
---
the divisorial Zariski decomposition
|  $\mathcal{C}_{p}$ and
---
the Alexandrov body construction
Section 5 | Morse inequality for $(\operatorname{\overline{Eff}}^{1}(X),\operatorname{vol})$ | Morse inequality for $(\mathcal{C}_{p},\operatorname{vol})$
Section 6 |  | movable curve classes
---
and duality of cones
| mixed area measures
---
and Minkowski’s existence theorem
Section 7 | addition of movable curve classes |  | Blaschke addition and
---
the Kneser-Süss inequality
Section 8 | fibration and restricted volumes | projection and projection volumes

In this dictionary, convex bodies with non-empty interior correspond to big
movable classes. We can phrase several classical convex geometry results due
to Minkowski and Alexandrov in terms of algebraic geometry; for example, the
divisorial Zariski decomposition is a variant of the Alexandrov body
construction.

We highlight a couple particular results. The following proposition111The
first part of this proposition has been implicitly contained in [FL13] and
[LX16b], and was also mentioned to the second author by S. Boucksom.
corresponds to the linearity of the $(n-1)$st mixed volume of polytopes.

###### Proposition 1.1

Let $X$ be a smooth projective variety of dimension $n$. Assume that $L$ is
pseudo-effective and $M$ is movable, then

$\langle L^{n-1}\rangle\cdot M=\langle L^{n-1}\cdot M\rangle.$

However, there exist projective manifolds such that $\langle L_{1}\cdot
L_{2}\cdot...\cdot L_{n-1}\rangle\cdot M\neq\langle L_{1}\cdot
L_{2}\cdot...\cdot L_{n-1}\cdot M\rangle$ if the pseudo-effective classes
$L_{i}$ are different.

We also discuss Brascamp-Lieb type inequalities on smooth projective
varieties; for example, we prove the following Loomis-Whitney inequality for
$(1,1)$ classes.

######  Theorem 1.2 (Theorem 8.1)

Let $X$ be a smooth projective variety of dimension $n$. Suppose that
$\\{f_{i}:X\dashrightarrow C_{i}\\}_{i=1}^{n}$ is a collection of dominant
rational maps onto curves, and assume that the induced map
$X\dashrightarrow\prod_{i=1}^{n}C_{i}$ is birational. Then for any big $(1,1)$
class $L$, we have

$\left(\operatorname{vol}(L)/n!\right)^{n-1}\leq\prod_{i=1}^{n}\left(\operatorname{vol}_{f_{i}}(L)/(n-1)!\right),$

where $\operatorname{vol}_{f_{i}}(L)$ is the restricted volume on a general
fiber of $f_{i}$ (up to a resolution). Moreover, if all the $f_{i}$ are
morphisms, then we obtain equality if and only if the positive part of $L$ is
numerically proportional to a sum of fibers of the $f_{i}$.

While our discussion of the fields of complex and convex geometry will be
purely by analogy, one can sometimes make the relationship between line
bundles and polytopes quite explicit. There is a well-understood way to
construct a polytope (up to translation) to a line bundle in the toric
setting. More generally, one has an analogous construction for arbitrary
varieties known as the Okounkov or Newton-Okounkov body (see e.g. [LM09],
[KK12]). Unfortunately, certain geometric features of Okounkov bodies rely
heavily on the choice of flag. Thus it is not clear whether the dictionary we
present here can be enriched to the polytope setting while still capturing the
full strength of our results. In particular, it is interesting to ask the
following question:

###### Question 1.3

Let $X$ be a compact Kähler manifold (or smooth projective variety) of
dimension $n$. After possibly fixing some data on $X$, is there a “canonical”
injective linear map

$i:H^{1,1}(X,\mathbb{R})\rightarrow
V:=C^{0}(\mathbb{S}^{n-1})/L(\mathbb{S}^{n-1}),$

or equivalently, surjective linear map

$i^{*}:V^{*}\rightarrow H^{n-1,n-1}(X,\mathbb{R}),$

such that the positive cones of $(1,1)$ classes are canonically embedded in
the positive cones of functions, the map $i$ preserves positivity and the
pairing between $H^{1,1}(X,\mathbb{R})$ and $H^{n-1,n-1}(X,\mathbb{R})$ is
induced by the pairing between $V$ and $V^{*}$ via the maps $i,i^{*}$?

### Acknowledgement

We would like to thank J. P. Demailly, M. Jonsson and V. Tosatti for several
comments and suggestions on this work, and R. Lazarsfeld, M. Mustaţă, and B.
Totaro for pointing out an error in an earlier version. The first author was
supported by an NSA Young Investigator Grant. The second author was supported
in part by CSC fellowship and Institut Fourier.

##  2\. Preliminaries

###  2.A. Positive classes

Let $X$ be a compact Kähler manifold of dimension $n$. We will let
$H^{1,1}(X,\mathbb{R})$ denote the real Bott-Chern cohomology group of
bidegree $(1,1)$. Note that the Bott-Chern cohomology coincides with de Rham
cohomology on a compact Kähler manifold, due to the
$\partial\bar{\partial}$-lemma. A Bott-Chern class is called pseudo-effective
if it contains a $d$-closed positive current. A pseudo-effective class $L$ is
called movable if for any irreducible divisor $Y$ the Lelong number $\nu(L,y)$
(or minimal multiplicity as in [Bou04]) vanishes at a very general point $y\in
Y$. We will be interested in the following cones in $H^{1,1}(X,\mathbb{R})$:

  * •

$\operatorname{\overline{Eff}}^{1}(X)$: the cone of pseudo-effective
$(1,1)$-classes;

  * •

$\operatorname{Mov}^{1}(X)$: the cone of movable $(1,1)$-classes.

Classes in the interior of $\operatorname{\overline{Eff}}^{1}(X)$ are known as
big classes; these are exactly the classes admitting a Kähler current (i.e.
strictly positive current), or equivalently, having strictly positive volumes.
As a consequence of Demailly’s regularization theorem [Dem92], any big class
contains a Kähler current with analytic singularities. Let
$H^{n-1,n-1}(X,\mathbb{R})$ denote the real Bott-Chern cohomology group of
bidegree $(n-1,n-1)$. We will be interested in the following cone in
$H^{n-1,n-1}(X,\mathbb{R})$:

  * •

$\operatorname{Mov}_{1}(X)$: the cone of movable $(n-1,n-1)$-classes.

Recall that $\operatorname{Mov}_{1}(X)$ is the closed cone generated by
classes of the form
$\mu_{*}(\widetilde{A}_{1}\cdot...\cdot\widetilde{A}_{n-1})$, where $\mu$ is a
modification and the $\widetilde{A}_{i}$ are Kähler classes upstairs, see e.g.
[BDPP13]. An irreducible curve $C$ is called movable if it is a member of an
algebraic family that covers $X$.

###### Remark 2.1

Let $\omega$ be a fixed Kähler class on $X$ and let $L$ be a $(1,1)$ class. By
[Bou04] the condition $L\in\operatorname{Mov}^{1}(X)$ is equivalent to
requiring that for any $\delta>0$ there is a modification
$\mu:\widehat{X}\rightarrow X$ such that
$L+\delta\omega=\mu_{*}\widehat{\omega}_{\delta}$, where
$\widehat{\omega}_{\delta}$ is a Kähler class on $\widehat{X}$. In particular,
when $n=2$, we have $\operatorname{Mov}^{1}(X)=\operatorname{Mov}_{1}(X)$.

The most important conjecture concerning these cones is Demailly’s conjecture
on the (weak) transcendental Morse inequality. This conjecture was recently
proved for projective manifolds in [WN16]. When working with positive cones,
it will at times be useful to restrict ourselves to this setting. In
particular, by [BDPP13, WN16], for projective manifolds we have

$\operatorname{\overline{Eff}}^{1}(X)^{*}=\operatorname{Mov}_{1}(X).$

###### Remark 2.2

By [Cut13] and [FL13, Section 2.2.2], the results of [BFJ09] can be extended
to an arbitrary algebraically closed field. Thus it is also possible to work
in the purely algebraic setting, i.e. with smooth algebraic varieties over an
arbitrary algebraically closed field (see also [LX16a, Section 2.3]). In this
case one should replace $H^{1,1}(X,\mathbb{R})$ by the space $N^{1}(X)$ of
$\mathbb{R}$-Cartier divisors up to numerical equivalence. All our results
hold equally well with this change.

###### Remark 2.3

As already noted in our previous works [LX16a, LX16b], the main results of
these papers extend to the transcendental situation whenever we have the
(weak) transcendental Morse inequality. In particular, the transcendental
versions hold on projective manifolds.

###  2.B. Positive functions on the sphere

Let $\mathfrak{K}^{n}$ denote the set of convex bodies (compact convex
subsets) with non-empty interior in $\mathbb{R}^{n}$. For two convex bodies
$K,L$, we have the Minkowski sum $K+L:=\\{x+y|\ x\in K,y\in L\\}$ and it is
clear $K+L\in\mathfrak{K}^{n}$. In this way, $(\mathfrak{K}^{n},+)$ is a
commutative semi-group. For $\lambda\geq 0$, the scaling operation is $\lambda
K:=\\{\lambda x|\ x\in K\\}$.

Associated with a convex body $K$ is its support function, $h(K,\cdot)$,
defined on $\mathbb{R}^{n}$ by

$\displaystyle h(K,x):=\max\\{x\cdot y|y\in K\\}.$

We shall write $h_{K}(\cdot)$, rather than $h(K,\cdot)$, for the support
function of $K$, and we will usually consider only the restriction of the
support function to $\mathbb{S}^{n-1}$. So $h_{K}$ is a continuous function on
$\mathbb{S}^{n-1}$. Let $C^{0}(\mathbb{S}^{n-1})$ denote the real valued
continuous function space on the sphere. Then the support functions induce a
map:

$\displaystyle\Phi:$ $\displaystyle\mathfrak{K}^{n}\rightarrow
C^{0}(\mathbb{S}^{n-1})$ $\displaystyle K\mapsto h_{K}(\cdot).$

Indeed, by the embedding theorem for spaces of convex sets (see e.g. [Hör55],
[Sch14, Chapter 1]), $\Phi$ is an isometrical isomorphism onto its image if we
endow the spaces $\mathfrak{K}^{n}$ and $C^{0}(\mathbb{S}^{n-1})$ with
suitable metrics. Let $d_{H}$ denote the Hausdorff metric on
$\mathfrak{K}^{n}$, that is,

$\displaystyle d_{H}(K,L)=\min\\{\epsilon>0|K\subseteq L+\epsilon B\ \&\
L\subseteq K+\epsilon B\\},$

where $B$ is the unit $n$-ball. We endow $C^{0}(\mathbb{S}^{n-1})$ with
maximal norm $|\cdot|_{\infty}$, so

$\displaystyle d_{H}(K,L)=|h_{K}-h_{L}|_{\infty}.$

Thus
$\Phi:(\mathfrak{K}^{n},d_{H})\rightarrow(\Phi(\mathfrak{K}^{n}),|\cdot|_{\infty})$
is an isometrical isomorphism. More importantly, by the above embedding
$\Phi$, we can identify $\mathfrak{K}^{n}$ with its image
$\Phi(\mathfrak{K}^{n})\subset C^{0}(\mathbb{S}^{n-1})$. Thus we will freely
pass back and forth between convex bodies and their support functions on the
sphere.

####  2.B.a. Quotient space

By the embedding theorem for convex bodies, $\mathfrak{K}^{n}$ can be seen as
a convex cone in an infinite dimensional Banach space. For
$a\in\mathbb{R}^{n}$ it is clear that $h_{K+a}=h_{K}+l_{a}$ where
$l_{a}(x)=a\cdot x$ is a linear function. Comparing the equality
characterization of Brunn-Minkowski inequalities in convex geometry with the
one in algebraic geometry (see Section 2.C), it is convenient to identify two
convex bodies if they differ by a translation. This leads to the quotient
space of $C^{0}(\mathbb{S}^{n-1})$ by linear functions. We denote
$L(\mathbb{S}^{n-1})$ to be the vector space whose elements are the
restriction of linear functions to $\mathbb{S}^{n-1}$. It is obvious that
$\dim L(\mathbb{S}^{n-1})=n$. We consider the quotient space

$V:=C^{0}(\mathbb{S}^{n-1})/L(\mathbb{S}^{n-1}).$

We endow $V$ with the quotient norm, that is, for $f\in V$ we define

$||f||=\inf_{l\in L(\mathbb{S}^{n-1})}|f-l|_{\infty}.$

It is a standard fact in functional analysis that $(V,||\cdot||)$ is also an
infinite dimensional Banach space.

###### Remark 2.4

Unlike the finite dimensional situation, the Banach space $V$ is not
reflexive. If we identify $V$ with its canonical image in $V^{**}$, then
$V\subsetneq V^{**}$. This means that not every continuous linear functional
on $V^{*}$ is given by a continuous function.

####  2.B.b. Positive cones on $V$

We will denote by $\mathcal{C}$ the image of $\mathfrak{K}^{n}$ in $V$, so
that $\mathcal{C}$ is a convex cone in $V$. We will also let
$\mathcal{C}_{p}\subset V$ denote the convex cone generated by non negative
continuous functions. Note that $\mathcal{C}\subset\mathcal{C}_{p}$. Motivated
by the toric case, we are guided by the following analogies:

$\operatorname{\overline{Eff}}^{1}(X)\leftrightsquigarrow\mathcal{C}_{p},\qquad\qquad\operatorname{Mov}^{1}(X)\leftrightsquigarrow\mathcal{C}.$

###  2.C. Volume and Brunn-Minkowski inequalities

In convex geometry, we have the following fundamental geometric inequality
(see e.g. [Sch14, Chapter 7]).

###### Theorem 2.5

Let $K,L\subseteq\mathbb{R}^{n}$ be two convex bodies with non-empty interior,
then

$\operatorname{vol}(K+L)^{1/n}\geq\operatorname{vol}(K)^{1/n}+\operatorname{vol}(L)^{1/n}$

with equality if and only if $K,L$ are homothetic.

In the complex geometry setting, the volume of a class
$L\in\operatorname{\overline{Eff}}^{1}(X)$ is defined to be

$\operatorname{vol}(L)=\sup_{T}\int_{X}T_{ac}^{n}$

for $T$ ranging over the closed positive $(1,1)$-currents in $L$, where
$T_{ac}$ denotes the absolutely continuous part (as in [Bou02]). In
particular, for divisor classes, this analytic definition coincides with the
usual one.

By extending a previous result of [BFJ09] from big nef classes to big movable
classes, we proved in our previous work [LX16b]:

###### Theorem 2.6

Let $X$ be a smooth projective variety of dimension $n$. Assume that $L,M$ are
two big and movable $(1,1)$-classes, then

$\operatorname{vol}(L+M)^{1/n}\geq\operatorname{vol}(L)^{1/n}+\operatorname{vol}(M)^{1/n}$

with equality if and if $L,M$ are proportional.

###### Remark 2.7

For non-projective Kähler manifolds, by [FX14] we also have a Brunn-Minkowski
inequality and its equality characterization for big and nef $(1,1)$ classes.

##  3\. Positive products and mixed volumes

For toric varieties, a big movable divisor corresponds to a convex polytope
(up to translation). The most important structure on the set of polytopes is
the mixed volume. [BFJ09] shows how, in the toric setting, the mixed volume
corresponds to the positive product construction defined below (up to a
constant factor).

###  3.A. Mixed volumes and area measures

In the convex geometry setting, let $K_{1},...,K_{r}\in\mathfrak{K}^{n}$ be
convex bodies. Then there is a polynomial relation

$\operatorname{vol}(t_{1}K_{1}+...+t_{r}K_{r})=\sum_{i_{1}+...+i_{r}=n}\frac{n!}{i_{1}!i_{2}!...i_{r}!}V(K_{1}^{i_{1}},...,K_{r}^{i_{r}})t_{1}^{i_{1}}\ldots
t_{r}^{i_{r}},$

where $t_{i}\in\mathbb{R}_{+}$ and the $V(K_{1}^{i_{1}},...,K_{r}^{i_{r}})$
are coefficients defining the mixed volumes. In particular, if
$K_{1}=K_{2}=...=K_{r}=K$ (up to translations), then the mixed volume is just
$\operatorname{vol}(K)$.

There is a variant of this construction that will frequently be useful for us.
The area measure $S(K,...,K;\cdot)$ of a convex body $K$ is, roughly speaking,
the pushforward of the $(n-1)$ dimensional Hausdorff measure on $\partial K$
by its Gauss map (see [Sch14, Chapter 4] for the precise definition).
Alternatively, the (mixed) area measures can be defined through mixed volumes
and Riesz representation theorem in a very simple way. We follow the elegant
discussion in [Lut88]. Suppose that $K_{1},...,K_{n-1}$ are fixed convex
bodies with non empty interior. Via the embedding map $\Phi$, the function

$l(h_{L}):=V(K_{1},...,K_{n-1},L)$

can be extended in a unique way to a continuous functional on
$C^{0}(\mathbb{S}^{n-1})$. Here, $V(K_{1},...,K_{n-1},L)$ is the mixed volume
of convex bodies (see [Sch14, Chapter 5]). Thus there exists a Borel measure,
denoted by $S(K_{1},...,K_{n-1};\cdot)$, such that

$\displaystyle
l(h_{L})=V(K_{1},...,K_{n-1},L)=\int_{\mathbb{S}^{n-1}}h_{L}(x)dS(K_{1},...,K_{n-1};x).$

The mixed area measure $S(K_{1},...,K_{n-1};\cdot)$ has the origin as its
centroid and does not concentrate on any great subsphere.

###  3.B. Positive products

Let $X$ be a compact Kähler manifold of dimension $n$. Before going on, we
first recall the definition of positive currents with minimal singularities.
Let $\\{\theta\\}$ be a pseudo-effective $(1,1)$ class, where $\theta$ is a
smooth representative. Then a positive current
$T=\theta+i\partial\bar{\partial}\varphi$ has minimal singularities if for any
$u\in\operatorname{PSH}(X,\theta)$ there exists some constant $c$ such that
$u\leq\varphi+c$. Note that such a current $T_{\min}$ always exists: let

$V_{\theta}:=\sup\\{\varphi\in\operatorname{PSH}(X,\theta)|\ \varphi\leq
0\\},$

then $\theta+dd^{c}V_{\theta}$ is a positive current with minimal
singularities. There may be many positive currents with minimal singularities
in a class.

Assume that $L_{1},...,L_{r}\in H^{1,1}(X,\mathbb{R})$ are big classes, i.e.
every class $L_{i}$ contains a Kähler current. By the theory developed in
[BEGZ10], we can associate to $L_{1},...,L_{r}$ a positive class in
$H^{r,r}(X,\mathbb{R})$, denoted by $\langle L_{1}\cdot...\cdot L_{r}\rangle$.
It is defined as the class of the non-pluripolar product of positive current
with minimal singularities, that is,

$\langle L_{1}\cdot...\cdot L_{r}\rangle:=\\{\langle T_{1,\min}\wedge...\wedge
T_{r,\min}\rangle\\}$

where $\langle T_{1,\min}\wedge...\wedge T_{r,\min}\rangle$ is the non-
pluripolar product. It is proved in [BEGZ10] that the positive product
$\langle L_{1}\cdot...\cdot L_{r}\rangle$ does not depend on the choices of
currents with minimal singularities. Moreover, if the $L_{i}$ are nef, then
$\langle L_{1}\cdot...\cdot L_{r}\rangle=L_{1}\cdot...\cdot L_{r}$ is the
usual cohomology intersection.

###### Remark 3.1

As proved in [Pri13, Proposition 1.10], it is also not hard to see that the
construction in [BDPP13, Theorem 3.5] and the construction in [BEGZ10] are the
same. This follows directly from the maximal integration property of $\langle
L_{1},...,L_{r}\rangle$ against smooth positive closed $(n-r,n-r)$-forms (see
[BEGZ10, Proposition 1.18 and Proposition 1.20]).

###### Remark 3.2

Let $X$ be a smooth projective variety of dimension $n$, and assume that
$L_{1},...,L_{r}$ are real big divisor classes. Then we can also define
$\langle L_{1},...,L_{r}\rangle$ by the algebraic construction in [BFJ09].
Since the positive product constructions in [BFJ09] are defined by taking
limits in the space $N_{n-r}(X)$ of $\mathbb{R}$-cycles up to numerical
equivalence (not in $H^{r,r}(X,\mathbb{R})$), it might be different from the
analytic construction, depending on the validity of the Hodge Conjectures.
However, the resulting positive products are the same for $r=1$ and $r=n-1$
because of the Lefschetz theorem on $(1,1)$ and $(n-1,n-1)$ classes. And the
equality is obvious for $r=n$.

In this section, we discuss the analogue between mixed volume (with Minkowski
addition) and positive products (with addition of divisors). We show that in
some ways this correspondence works well, but in other ways it fails. The next
proposition corresponds to the fact that the function $V(K^{n-1},-)$ is linear
on $\mathfrak{K}^{n}$.

###### Proposition 3.3

Let $X$ be a smooth projective variety of dimension $n$. Assume that $L$ is
pseudo-effective and $M$ is movable, then

$\langle L^{n-1}\rangle\cdot M=\langle L^{n-1}\cdot M\rangle.$

  * Proof.

This is an application of [FL13, Lemma 6.21]. For completeness, we include
some arguments here. Firstly, by [WN16] the results in [BFJ09] can extend to
transcendental classes.

By taking limits, we can assume that $M=\pi_{*}A$ for some birational
modification $\pi$ and Kähler class $A$ upstairs. We claim that
$\pi^{*}\langle L^{n-1}\rangle=\langle(\pi^{*}L)^{n-1}\rangle$. Note that
$\pi_{*}\langle(\pi^{*}L)^{n-1}\rangle=\langle L^{n-1}\rangle$, so we need to
check that $\langle(\pi^{*}L)^{n-1}\rangle\cdot E=0$ for any irreducible
$\pi$-exceptional divisor $E$. For any $c>0$, the restriction
$\pi^{*}L+cE|_{E}$ can not be pseudo-effective, thus $\pi^{*}L$ can not be
$E$-big. By [BFJ09, Lemma 4.10], we get that
$\langle(\pi^{*}L)^{n-1}\rangle\cdot E=0$, concluding the proof of our claim.
Since $\pi^{*}(\pi_{*}A)\succeq A$, by basic properties of positive products,
we have

$\displaystyle\langle L^{n-1}\cdot\pi_{*}A\rangle$
$\displaystyle=\langle(\pi^{*}L)^{n-1}\cdot\pi^{*}(\pi_{*}A)\rangle$
$\displaystyle\geq\langle(\pi^{*}L)^{n-1}\cdot
A\rangle=\langle(\pi^{*}L)^{n-1}\rangle\cdot A$ $\displaystyle=\pi^{*}\langle
L^{n-1}\rangle\cdot A=\langle L^{n-1}\rangle\cdot\pi_{*}A$

On the other hand, we always have $\langle L^{n-1}\rangle\cdot M\geq\langle
L^{n-1}\cdot M\rangle$. Thus we have the equality $\langle L^{n-1}\rangle\cdot
M=\langle L^{n-1}\cdot M\rangle$. $\Box$

###### Remark 3.4

In the above proof, the key ingredient is the equality $\pi^{*}\langle
L^{n-1}\rangle=\langle(\pi^{*}L)^{n-1}\rangle$. An alternative way to see this
equality is to use the function $\mathfrak{M}$ studied in [Xia15a, LX16b]. For
any big class $N$, we have

$\displaystyle\frac{\pi^{*}\langle L^{n-1}\rangle\cdot
N}{\operatorname{vol}(N)^{1/n}}$ $\displaystyle\geq\frac{\langle
L^{n-1}\rangle\cdot\pi_{*}N}{\operatorname{vol}(\pi_{*}N)^{1/n}}.$

Thus the infimum of LHS is obtained by some big class $N$ satisfying
$\pi_{*}N=L$, which implies that the infimum of LHS is obtained by the
positive part of $\pi^{*}L$. By [LX16b] we get that

$\pi^{*}\langle L^{n-1}\rangle=\langle
P(\pi^{*}L)^{n-1}\rangle=\langle(\pi^{*}L)^{n-1}\rangle.$

###### Corollary 3.5

Let $X$ be a smooth projective variety of dimension $n$. Assume that $L$ is
pseudo-effective and $M_{1},M_{2}$ are movable, then

$\langle L^{n-1}\cdot(M_{1}+M_{2})\rangle=\langle L^{n-1}\cdot
M_{1}\rangle+\langle L^{n-1}\cdot M_{2}\rangle.$

However, the more general statement $\langle L_{1}\cdot L_{2}\cdot...\cdot
L_{n-1}\rangle\cdot M=\langle L_{1}\cdot L_{2}\cdot...\cdot L_{n-1}\cdot
M\rangle$ is false. Equivalently, it is not true that $\phi^{*}\langle
L_{1}\cdot L_{2}\cdot...\cdot
L_{n-1}\rangle=\langle\phi^{*}L_{1}\cdot\phi^{*}L_{2}\cdot...\cdot\phi^{*}L_{n-1}\rangle$
for a birational map $\phi$. If this property were true, it would easily imply
that the volume function is polynomial (not just locally polynomial)
everywhere in the interior of the movable cone. But this property fails in the
following examples suggested by Lazarsfeld and Totaro.

######  Example 3.6 (A flop)

Let $X$ be the projective bundle (of quotients) over $\mathbb{P}^{1}$ defined
by $\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-1)$. There are two natural
divisor classes on $X$: the class $f$ of the fibers of the projective bundle
and the class $\xi$ of the sheaf $\mathcal{O}_{X/\mathbb{P}^{1}}(1)$. Using
for example [Ful11] and [FL13], one sees that $f$ and $\xi$ generate the
algebraic cohomology classes with the relations $f^{2}=0$,
$\xi^{2}f=-\xi^{3}=1$ and

$\operatorname{\overline{Eff}}^{1}(X)=\operatorname{Mov}^{1}(X)=\langle
f,\xi\rangle\qquad\qquad\operatorname{Nef}^{1}(X)=\langle f,\xi+f\rangle$

and

$\displaystyle\operatorname{\overline{Eff}}_{1}(X)=\langle\xi
f,\xi^{2}\rangle\qquad$
$\displaystyle\qquad\operatorname{Nef}_{1}(X)=\langle\xi f,\xi^{2}+\xi
f\rangle.$

We can calculate volume by pushing forward. We have if $b\geq a$,

$\displaystyle H^{0}(X,a\xi+bf)$
$\displaystyle=H^{0}(\mathbb{P}^{1},\operatorname{Sym}^{a}(\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-1))\otimes\mathcal{O}(b))$
$\displaystyle=(b+1)(a+1)+ba+(b-1)(a-1)+\ldots+(b-a)(a-a)$
$\displaystyle=ba^{2}/2-a^{3}/6+3ab/2+b+7a/6+1$

and if $b\leq a$,

$\displaystyle H^{0}(X,a\xi+bf)$
$\displaystyle=(b+1)(a+1)+ba+(b-1)(a-1)+\ldots+(b-b)(a-b)$
$\displaystyle=ab^{2}/2-b^{3}/6+3ba/2+a+7b/6+1$

So for $b\geq a$ the volume is $3ba^{2}-a^{3}$, and for $a\geq b$ it is
$3ab^{2}-b^{3}$. Note that in a neighborhood of $a=b$ the volume is not
locally polynomial.

######  Example 3.7 (Projective bundle over an abelian surface)

In fact, the volume function can even be transcendental on certain regions of
the big movable cone. [KLM13] gives an example of a $\mathbb{P}^{2}$-bundle
over an abelian twofold whose volume function is transcendental on a region of
the big cone of divisors. While this region only contains non-movable
divisors, their computation also extends into the movable cone as follows.

As in [KLM13, Section 3] let $A$ be the self-product of an elliptic curve
without CM and let $f_{1}$, $f_{2}$, and $\Delta$ denote the fibers of the two
projection maps and the diagonal. Consider the projective bundle $\pi:X\to A$
defined as

$X=\mathbb{P}_{A}(\mathcal{O}(f_{1}+f_{2}+\Delta)\oplus\mathcal{O}(-f_{1})\oplus\mathcal{O}(-f_{2}))$

and let $L$ be a divisor representing $\mathcal{O}_{X/A}(1)$.

For positive real numbers $c_{1},c_{2},c_{3}$ [KLM13, Section 3] computes the
volume of $L+\pi^{*}(c_{1}f_{1}+c_{2}f_{2}+c_{3}\Delta)$ as an explicit
integral over a region in the plane. This divisor is movable but not nef when
$c_{1}=\frac{1}{2}+\epsilon,c_{2}=\frac{1}{2}+\epsilon,c_{3}=1$ for $\epsilon$
sufficiently small (for example, $\epsilon=1/10$ will work). Following the
lengthy computation of [KLM13], we find that for rational values of the
$c_{i}$ in a neighborhood of the divisor
$L+\frac{6}{10}\pi^{*}f_{1}+\frac{6}{10}\pi^{*}f_{2}+\pi^{*}\Delta$ the volume
can be written as a (non-zero) quadratic irrational multiple of $\mathrm{ln}$
of a (non-constant) quadratic irrational, plus an algebraic number.

###### Remark 3.8

In the algebraic setting, for any big and movable classes
$L_{1},\ldots,L_{n-1}$ the difference $\phi^{*}\langle L_{1}\cdot
L_{2}\cdot...\cdot
L_{n-1}\rangle-\langle\phi^{*}L_{1}\cdot\phi^{*}L_{2}\cdot...\cdot\phi^{*}L_{n-1}\rangle$
is a pseudo-effective $\phi$-contracted curve class. This follows from the
argument establishing negativity of contraction.

##  4\. Alexandrov body and Zariski decompositions

In this section, we interpret a classical construction due to Alexandrov
[Ale38] (an English translation is available in his selected works [Ale96]) as
a Zariski decomposition structure in convex geometry. This is the convex
geometry version of the $\sigma$-decomposition (or divisorial Zariski
decomposition) of divisors in algebraic geometry.

###  4.A. Zariski decomposition

First we recall the notion of a $\sigma$-decomposition. Let $X$ be a smooth
projective variety of dimension $n$, and let
$L\in\operatorname{\overline{Eff}}^{1}(X)$ be a big divisor class. As in the
theories developed in [Bou04, Nak04, BFJ09, BDPP13], there exists a
decomposition

$L=P(L)+N(L)$

where $P(L)=\langle L\rangle$ and $N(L)=L-P(L)$. This decomposition is known
as the $\sigma$-decomposition; it should be interpreted as replacing a pseudo-
effective class $L$ with a movable class $P(L)\leq L$ with no “loss” of
positivity. The key properties of this decomposition are the following:

  1. 1.

$P(L)\in\operatorname{Mov}^{1}(X)$ and $N(L)$ is a rigidly embedded effective
divisor class222Here we use the terminology of [Bou04], “rigidly embedded”
means that the class $N(L)$ contains only one positive current..

  2. 2.

$\operatorname{vol}(L)=\operatorname{vol}(P(L))$.

  3. 3.

The orthogonality property: $\langle P(L)^{n-1}\rangle\cdot N(L)=0$.

The first two properties characterize the $\sigma$-decomposition: if $L=P+N$
is any decomposition such that
$P\in\operatorname{Mov}^{1}(X),N\in\operatorname{\overline{Eff}}^{1}(X)$ and
$\operatorname{vol}(L)=\operatorname{vol}(P)$, then it is shown in [FL13,
Section 5] that we must have $P=P(L),N=N(L)$.

In fact, [LX16b] shows a stronger property: the $\sigma$-decomposition exactly
captures the failure of the volume function to be strictly log concave.

######  Theorem 4.1 (see [LX16b] Theorem 1.6)

Let $X$ be a smooth projective variety of dimension $n$. Let $L$ and $D$ be
big divisor classes on $X$. Then we have

$\operatorname{vol}(L+D)^{1/n}=\operatorname{vol}(L)^{1/n}+\operatorname{vol}(D)^{1/n}$

if and only if $P(L),P(D)$ are proportional.

###  4.B. Alexandrov body

Let us recall the construction of the Alexandrov body (see e.g. [Sch14,
Section 7.5]). Let $\mathfrak{K}_{0}^{n}\subseteq\mathfrak{K}^{n}$ be the
space of convex bodies with $0$ as an interior point. Let $f$ be a strictly
positive continuous function. The Alexandrov body associated with $f$ is
defined as the unique maximal element (denoted by $K$), with respect to set
inclusion, of the set

$\\{Q\in\mathfrak{K}_{0}^{n}|\ h_{Q}\leq f\\}.$

Equivalently, this means that the support function $h_{K}=\sup\rho$ where the
sup is taken over all 1-homogeneous convex functions less than or equal to
$f$. The convex body $K$ can also be characterized as follows:

$K=\bigcap_{y\in\mathbb{S}^{n-1}}\\{x\in\mathbb{R}^{n}|\ x\cdot y\leq
f(y)\\}.$

If we denote $P(f):=h_{K}$, then we have

$P(f)=f,\ a.e.\ \textrm{with respect to}\ \mu_{K},$ (4.1)

where $\mu_{K}:=S(K^{n-1};\cdot)$ is the area measure of $K$. Equivalently, we
get

$(f-P(f))\cdot\mu_{K}:=\int_{\mathbb{S}^{n-1}}(f-P(f))d\mu_{K}=0.$ (4.2)

The volume of $f$ is defined to be

$\operatorname{vol}(f):=\operatorname{vol}(K).$ (4.3)

For $L\in\mathfrak{K}_{0}^{n}$, it is clear that
$P(h_{L})=h_{L},\operatorname{vol}(h_{L})=\operatorname{vol}(L)$.

####  4.B.a. Decomposition theory

As an immediate consequence of the characterization of Alexandrov body, we
have

###### Lemma 4.2

Let $f,g$ be two strictly positive continuous functions such that $f=g+l_{a}$
for some linear function $l_{a}$, then we have $P(f)=P(g)+l_{a}$.

This implies the decomposition of $\mathcal{C}_{p}$:

######  Theorem 4.3 (Decomposition of $(\mathcal{C}_{p},\operatorname{vol})$
[Ale38])

Assume that $f\in\mathcal{C}_{p}$ is represented by a strictly positive
continuous function. Then there exists a decomposition

$f=P(f)+N(f)$

satisfying:

  * •

$P(f)\in\mathcal{C}$ and $N(f)=f-P(f)\in\mathcal{C}_{p}$;

  * •

$\operatorname{vol}(f)=\operatorname{vol}(P(f))$;

  * •

if we assume $P(f)$ is represented by a convex body $K$ and denote $\langle
P(f)^{n-1}\rangle:=S(K^{n-1};\cdot)$, then the orthogonality estimate holds in
the sense that $\langle P(f)^{n-1}\rangle\cdot N(f)=0$.

We can rephrase this result to look exactly analogous to the characterization
of the $\sigma$-decomposition described above; by the Brunn-Minkowski theorem
we obtain

###### Proposition 4.4

Suppose that $f,g\in\mathcal{C}_{p}$ are represented by strictly positive
continuous functions. Then we have

$\operatorname{vol}(f+g)^{1/n}=\operatorname{vol}(f)^{1/n}+\operatorname{vol}(g)^{1/n}$

if and only if $P(f),P(g)$ are proportional. In particular, if $f=P+N$ is a
decomposition such that $P\in\mathcal{C},N\in\mathcal{C}_{p}$ and
$\operatorname{vol}(P)=\operatorname{vol}(f)$, then we must have
$P=P(f),N=N(f)$.

  * Proof.

The proof is similar to the complex geometry setting. We only need to prove
the second point. Integrating $f=P+N$ against with the area measure $\langle
P(f)^{n-1}\rangle$ implies

$\displaystyle\operatorname{vol}(f)$ $\displaystyle=P\cdot\langle
P(f)^{n-1}\rangle+N\cdot\langle P(f)^{n-1}\rangle$ $\displaystyle\geq
P\cdot\langle
P(f)^{n-1}\rangle\geq\operatorname{vol}(P)^{1/n}\operatorname{vol}(P(f))^{n-1/n}$
$\displaystyle=\operatorname{vol}(f),$

where the first inequality follows from $N\in\mathcal{C}_{p}$, the second
inequality follows from the Brunn-Minkowski inequality and the last equality
follows from the assumption $\operatorname{vol}(P)=\operatorname{vol}(f)$.
Thus we get equality everywhere, in particular, $P\cdot\langle
P(f)^{n-1}\rangle=\operatorname{vol}(P)^{1/n}\operatorname{vol}(P(f))^{n-1/n}$.
Note that $P\cdot\langle P(f)^{n-1}\rangle$ is just the mixed volume computed
by the convex bodies representing $P$ and $P(f)$. By Brunn-Minkowski theory,
up to some translation these two convex bodies are equal, which in turn
implies $P=P(f)$. $\Box$

###### Remark 4.5

Theorem 4.3 can also be seen as some kind of infinite dimensional extension of
the theory developed in [LX16a] when applied to this particular setting.

###### Remark 4.6

One should compare the Alexandrov body construction with the Berman-Boucksom
orthogonality estimate in complex pluripotential theory (see [BB10]), where
the authors apply a complex envelope construction analogous to the
construction of $P(f)$. This complex envelope construction admits similar
orthogonality properties as the Zariski decomposition, and has played an
important role in the variational approach to solve complex Monge-Ampère
equations (see [BBGZ13]). Indeed, by the theory developed in [BFJ15] on non-
Archimedean Monge-Ampère equations, the orthogonality in the divisorial
Zariski decomposition is closely related to non-Archimedean Berman-Boucksom
orthogonality.

##  5\. Morse type inequalities

Let $X$ be a compact Kähler manifold of dimension $n$, and let $A,B\in
H^{1,1}(X,\mathbb{R})$ be nef classes. Demailly’s conjecture on the (weak)
transcendental Morse inequality says that

$\operatorname{vol}(A-B)\geq A^{n}-nA^{n-1}\cdot B.$

This important conjecture was recently proved for projective manifolds in
[WN16]. For non-projective Kähler manifolds, it is proved in [Pop16] that
$A^{n}-nA^{n-1}\cdot B>0$ implies $A-B$ is big (see also [Xia15b] for a weaker
result).

As an easy corollary of the main result of [WN16], we have

###### Proposition 5.1

Let $X$ be a smooth projective variety of dimension $n$. If $L,M\in
H^{1,1}(X,\mathbb{R})$ are movable classes, then

$\operatorname{vol}(L-M)\geq\operatorname{vol}(L)-n\langle L^{n-1}\rangle\cdot
M.$

  * Proof.

For reader’s convenience, we repeat some arguments from [Xia14]. Without loss
of generality, we can assume that $L$ is big and that
$M=\mu_{*}\widehat{\omega}$ for a birational modification $\mu$ and for some
Kähler class $\widehat{\omega}$ upstairs. Take a suitable sequence of Fujita
approximations, $\pi_{m}^{*}(\mu^{*}L)=\omega_{m}+[E_{m}]$, such that

$\operatorname{vol}(\mu^{*}L)=\lim_{m\rightarrow\infty}\omega_{m}^{n},\ \ \
\langle(\mu^{*}L)^{n-1}\rangle\cdot\widehat{\omega}=\lim_{m\rightarrow\infty}\omega_{m}^{n-1}\cdot\pi_{m}^{*}\widehat{\omega}.$

By [WN16], we always have
$\operatorname{vol}(\omega_{m}-\pi_{m}^{*}\widehat{\omega})\geq\omega_{m}^{n}-n\omega_{m}^{n-1}\cdot\pi_{m}^{*}\widehat{\omega}$,
which implies that

$\displaystyle\operatorname{vol}(\mu^{*}L-\widehat{\omega})$
$\displaystyle=\operatorname{vol}(\pi_{m}^{*}(\mu^{*}L-\widehat{\omega}))$
$\displaystyle\geq\operatorname{vol}(\omega_{m}-\pi_{m}^{*}\widehat{\omega})$
$\displaystyle\geq\omega_{m}^{n}-n\omega_{m}^{n-1}\cdot\pi_{m}^{*}\widehat{\omega}.$

Let $m$ tend to infinity, we get
$\operatorname{vol}(\mu^{*}L-\widehat{\omega})\geq\operatorname{vol}(L)-n\langle(\mu^{*}L)^{n-1}\rangle\cdot\widehat{\omega}$.
This implies the desired inequality

$\operatorname{vol}(L-\mu_{*}\widehat{\omega})\geq\operatorname{vol}(L)-n\langle
L^{n-1}\rangle\cdot\mu_{*}\widehat{\omega}.$

$\Box$

###  5.A. Convex geometry analogue

Due to the importance of the Morse inequality in complex geometry, it is
interesting to ask for an analogue for convex bodies. More precisely, let
$K,L\in\mathfrak{K}^{n}$ and let $h_{K},h_{L}\in\mathcal{C}$ be their support
functions, one may wonder if

$\operatorname{vol}(h_{K}-h_{L})\geq\operatorname{vol}(K)-nV(K^{n-1},L)$

holds, and in particular, $h_{K}-h_{L}$ could be represented by a strictly
positive function if $\operatorname{vol}(K)-nV(K^{n-1},L)>0$.

Note that we can assume $\operatorname{vol}(K)-nV(K^{n-1},L)>0$, since the
inequality is trivial otherwise. It is also easy to see that the inequality
holds if $K,L$ are proportional up to some translation, so we may assume
otherwise.

We first verify that when $\operatorname{vol}(K)-nV(K^{n-1},L)>0$, the
function $h_{K}-h_{L}$ can be represented by a strictly positive continuous
function. This can be accomplished by applying the Diskant inequality (see
[Sch14, Section 7.2]). Recall that the _inradius_ of $K$ relative to $L$ is
defined by:

$r(K,L):=\max\\{\lambda>0|\lambda L+t\subset K\ \textrm{for some}\
t\in\mathbb{R}^{n}\\}.$

Applying the Diskant inequality to $K,L$, we get

$\displaystyle r(K,L)$
$\displaystyle\geq\frac{V(K^{n-1},L)^{1/n-1}-\left(V(K^{n-1},L)^{n/n-1}-\operatorname{vol}(K)\operatorname{vol}(L)^{1/n-1}\right)^{1/n}}{\operatorname{vol}(L)^{1/n-1}}.$

The generalized binomial formula yields

$\displaystyle
V(K^{n-1},L)^{1/n-1}-\left(V(K^{n-1},L)^{n/n-1}-\operatorname{vol}(K)\operatorname{vol}(L)^{1/n-1}\right)^{1/n}$
$\displaystyle=\sum_{k=1}^{\infty}\left(\begin{array}[]{c}1/n\\\
k\end{array}\right)(-1)^{k+1}\operatorname{vol}(K)^{k}\operatorname{vol}(L)^{k/n-1}V(K^{n-1},L)^{\frac{1-kn}{n-1}}$
$\displaystyle=\frac{1}{n}\operatorname{vol}(K)\operatorname{vol}(L)^{1/n-1}V(K^{n-1},L)^{-1}+\sum_{k=2}^{\infty}...$
$\displaystyle\geq\frac{1}{n}\operatorname{vol}(K)\operatorname{vol}(L)^{1/n-1}V(K^{n-1},L)^{-1},$

because every term in the sum $\sum_{k=2}^{\infty}...$ is non negative.
Substituting, we obtain

$r(K,L)\geq\frac{\operatorname{vol}(K)}{nV(K^{n-1},L)}.$

Now the assumption $\operatorname{vol}(K)-nV(K^{n-1},L)>0$ implies that, up to
a translation, $K\supset(1+\delta)L$ for some $\delta>0$. In particular, up to
a linear function, we have $h_{K}-h_{L}\geq\delta h_{L}$. This implies that
the class $\\{h_{K}-h_{L}\\}\in\mathcal{C}_{p}$ can be represented by a
strictly positive continuous function.

###### Remark 5.2

It is of course not true in general that $h_{K}-h_{L}$ can be represented by a
strictly positive convex continuous function. For example, this necessarily
fails when $K$ is indecomposable as discussed in Section 5.A.a.

Now the Morse inequality for convex bodies follows from the following result
due to [Ale38] (see also [Sch14, Section 7.5]).

###### Lemma 5.3

Let $f\in\mathcal{C}_{p}$ be a strictly positive continuous function with its
Alexandrov body given by $P(f)=h_{Q}$. For any continuous function $g\in V$ we
have

$\left.\frac{d}{dt}\right|_{t=0}\operatorname{vol}(f+tg)=n\langle
P(f)^{n-1}\rangle\cdot g.$

Applying this result to our original set-up, we see that for $t\in[0,1]$

$\displaystyle\operatorname{vol}(h_{K}-h_{L})-\operatorname{vol}(K)$
$\displaystyle=\int_{0}^{1}\frac{d}{dt}\operatorname{vol}(h_{K}-th_{L})dt$
$\displaystyle=\int_{0}^{1}-n\langle P(h_{K}-th_{L})^{n-1}\rangle\cdot
h_{L}dt$ $\displaystyle\geq-nV(K^{n-1},L),$

where the last inequality follows from the monotone property of mixed volumes.

Summarizing the discussions above, we get

######  Theorem 5.4 (Morse inequality for convex bodies)

Let $h_{K},h_{L}\in\mathcal{C}$. If

$\operatorname{vol}(K)-nV(K^{n-1},L)>0,$

then $h_{K}-h_{L}\in\mathcal{C}_{p}^{\circ}$ which means that it can be
represented by some strictly positive continuous function. Moreover, we have
$\operatorname{vol}(h_{K}-h_{L})\geq\operatorname{vol}(K)-nV(K^{n-1},L)$.

For convex bodies, we also have the “reverse” Khovanskii-Teissier inequality
(see [LX16a, Remark 4.18] for this terminology).

###### Proposition 5.5

Fix three convex bodies $K,L,M\in\mathfrak{K}^{n}$, then

$\displaystyle
nV(L,K^{n-1})V(K,M^{n-1})\geq\operatorname{vol}(K)V(L,M^{n-1}).$

  * Proof.

By taking limits, we can assume the convex bodies have non-empty interior.
After scaling, we can assume $V(K,M^{n-1})=V(L,M^{n-1})$. Under this
assumption, we need to show $nV(L,K^{n-1})\geq\operatorname{vol}(K)$.
Otherwise we would have $h_{K}-h_{L}\in\mathcal{C}_{p}^{\circ}$ by Theorem
5.4. Thus

$\int_{\mathbb{S}^{n-1}}(h_{K}-h_{L})dS(M^{n-1})=V(K,M^{n-1})-V(L,M^{n-1})>0,$

a contradiction. $\Box$

####  5.A.a. Indecomposability

It is interesting to ask when a big movable $(1,1)$-class lies on the boundary
(or extremal ray) of the movable cone. The analogue in the convex geometry
setting is the notion of indecomposability. Recall that $K$ is _decomposable_
if there exist convex bodies $M,D$ not homothetic to $K$ such that $K=M+D$.
Otherwise, we call that $K$ is _indecomposable_. In particular, the existence
of indecomposable convex bodies implies that, in the above Morse type
inequality for convex bodies, in general $h_{K}-h_{L}$ cannot be represented
by a support function. We refer the readers to [Sch14, Chapter 3] for
decomposable and indecomposable convex bodies.

###### Example 5.6

Suppose that $X$ is a toric variety defined by a fan $\Sigma$. Suppose that
$P$ is an indecomposable polytope whose normal fan is refined by $\Sigma$.
Then $P$ induces a big movable divisor on the boundary of the movable cone.
The converse is false – however, if one focuses only on integer multiples of
lattice polytopes, there is a good converse statement as in [CLS11,
Proposition 6.2.13].

For example, the only two-dimensional indecomposable convex bodies are
triangles. In the toric setting, a triangle represents the pullback under a
birational contraction of an ample generator of a ($\mathbb{Q}$-factorial)
toric surface of Picard rank $1$. Such a divisor is necessarily on the
boundary of the movable cone.

###  5.B. Reverse Khovanskii-Teissier inequality

In the algebraic setting, an important step in the analysis of the Morse type
bigness criterion is the “reverse” Khovanskii-Teissier inequality for big and
nef divisors $A$, $B$, and a movable curve class $\beta$:

$n(A\cdot B^{n-1})(B\cdot\beta)\geq B^{n}(A\cdot\beta).$

We prove a more general statement on “reverse” Khovanskii-Teissier
inequalities in the analytic setting333Indeed, this kind of generalization has
already been noted in [Xia15b, Remark 3.1], and it was also noted that the
constant ${k!(n-k)!}/{4n!}$ there could be improved by combining with the
technique of [Pop16]. Some related work has also appeared independently in the
recent preprint [Pop15]..

###### Theorem 5.7

Let $X$ be a compact Kähler manifold of dimension $n$. Let
$\omega,\beta,\gamma$ be three nef $(1,1)$ classes on $X$. Then we have

$(\beta^{k}\cdot\alpha^{n-k})\cdot(\alpha^{k}\cdot\gamma^{n-k})\geq\frac{k!(n-k)!}{n!}\alpha^{n}\cdot(\beta^{k}\cdot\gamma^{n-k}).$

  * Proof.

The proof depends on solving Monge-Ampère equations and the method of [Pop16].
Without loss of generality, we can assume that $\gamma$ is normalised such
that $\beta^{k}\cdot\gamma^{n-k}=1$. Then we need to show

$\displaystyle(\beta^{k}\cdot\alpha^{n-k})\cdot(\alpha^{k}\cdot\gamma^{n-k})\geq\frac{k!(n-k)!}{n!}\alpha^{n}.$
(5.4)

We first assume $\alpha,\beta,\gamma$ are all Kähler classes. We will use the
same symbols to denote the Kähler metrics in corresponding Kähler classes. By
the Calabi-Yau theorem [Yau78], we can solve the following Monge-Ampère
equation:

$\displaystyle(\alpha+i\partial\bar{\partial}\psi)^{n}=\left(\int\alpha^{n}\right)\beta^{k}\wedge\gamma^{n-k}.$
(5.5)

Denote by $\alpha_{\psi}$ the Kähler metric
$\alpha+i\partial\bar{\partial}\psi$. Then we have

$\displaystyle(\beta^{k}\cdot\alpha^{n-k})\cdot(\alpha^{k}\cdot\gamma^{n-k})$
$\displaystyle=\int\beta^{k}\wedge\alpha_{\psi}^{n-k}\cdot\int\alpha_{\psi}^{k}\wedge\gamma^{n-k}$
$\displaystyle=\int\frac{\beta^{k}\wedge\alpha_{\psi}^{n-k}}{\alpha_{\psi}^{n}}\alpha_{\psi}^{n}\cdot\int\frac{\alpha_{\psi}^{k}\wedge\gamma^{n-k}}{\alpha_{\psi}^{n}}\alpha_{\psi}^{n}$
$\displaystyle\geq\left(\int\left(\frac{\beta^{k}\wedge\alpha_{\psi}^{n-k}}{\alpha_{\psi}^{n}}\cdot\frac{\alpha_{\psi}^{k}\wedge\gamma^{n-k}}{\alpha_{\psi}^{n}}\right)^{1/2}\alpha_{\psi}^{n}\right)^{2}.$

The last line follows because of the Cauchy-Schwarz inequality. We claim that
the following pointwise inequality holds:

$\displaystyle\frac{\beta^{k}\wedge\alpha_{\psi}^{n-k}}{\alpha_{\psi}^{n}}\cdot\frac{\alpha_{\psi}^{k}\wedge\gamma^{n-k}}{\beta^{k}\wedge\gamma^{n-k}}\geq\frac{k!(n-k)!}{n!}.$
(5.6)

Then by (5.5) it is clear that the above pointwise inequality implies the
desired inequality (5.4). Now we prove the claim. For any fixed point $p\in
X$, we can choose some coordinates such that at the point $p$:

$\alpha_{\psi}=i\sum_{j=1}^{n}dz^{j}\wedge
d\bar{z}^{j},\quad\beta=i\sum_{j=1}^{n}\mu_{j}dz^{j}\wedge d\bar{z}^{j},$

and

$\gamma^{n-k}=i^{n-k}\sum_{|I|=|J|=n-k}\Gamma_{IJ}dz_{I}\wedge d\bar{z}_{J}.$

Denote by $\mu_{J}$ the product $\mu_{j_{1}}...\mu_{j_{k}}$ with index
$J=(j_{1}<...<j_{k})$ and denote by $J^{c}$ the complement index of $J$. Then
it is easy to see at the point $p$ we have

$\frac{\beta^{k}\wedge\alpha_{\psi}^{n-k}}{\alpha_{\psi}^{n}}\cdot\frac{\alpha_{\psi}^{k}\wedge\gamma^{n-k}}{\beta^{k}\wedge\gamma^{n-k}}=\frac{k!(n-k)!}{n!}\frac{(\sum_{J}\mu_{J})(\sum_{K}\Gamma_{KK})}{\sum_{J}\mu_{J}\Gamma_{J^{c}J^{c}}}\geq\frac{k!(n-k)!}{n!}.$

This then finishes the proof of the case when $\alpha,\beta,\gamma$ are all
Kähler classes. If they are just nef classes, by taking limits, then we get
the desired inequality. $\Box$

By taking suitable Fujita approximations, we immediately get

###### Corollary 5.8

Let $X$ be a compact Kähler manifold of dimension $n$. Let
$\alpha,\beta,\gamma$ be three pseudo-effective $(1,1)$ classes on $X$. Then
we have

$\langle\beta^{k}\cdot\alpha^{n-k}\rangle\cdot\langle\alpha^{k}\cdot\gamma^{n-k}\rangle\geq\frac{k!(n-k)!}{n!}\operatorname{vol}(\alpha)\langle\beta^{k}\cdot\gamma^{n-k}\rangle.$

  * Proof.

By the continuity of positive products, we can assume that the classes are
big. We may take a sequence of Fujita approximations $\mu_{m}:X_{m}\rightarrow
X$ such that

$\mu_{m}^{*}\alpha=\alpha_{m}+[E_{m}],\ \ \mu_{m}^{*}\beta=\beta_{m}+[F_{m}],\
\ \mu_{m}^{*}\gamma=\gamma_{m}+[G_{m}],$

where $\alpha_{m},\beta_{m},\gamma_{m}$ are Kähler and $E_{m},F_{m},G_{m}$ are
effective. Moreover, the positive products are the limits of intersections of
Kähler classes upstairs, for instance,

$\langle\beta^{k}\cdot\alpha^{n-k}\rangle=\lim_{m\rightarrow\infty}\beta_{m}^{k}\cdot\alpha_{m}^{n-k}.$

Then the inequality follows from Theorem 5.7 by taking limits. $\Box$

In the convex geometry setting, it is natural to ask whether an analogue holds
for convex bodies. As the generalization of Proposition 5.5, we get the
following result.

###### Theorem 5.9

Assume that $K,L,M$ are three convex bodies in $\mathbb{R}^{n}$. Then

$\displaystyle
V(K^{k},L^{n-k})V(L^{k},M^{n-k})\geq\frac{k!(n-k)!}{n!}\operatorname{vol}(L)V(K^{k},M^{n-k}).$

Since we did not find a similar statement in the literature, we provide a
brief proof below. The proof is inspired by [ADM99], where the authors
reproved some of the Alexandov-Fenchel inequalities by using mass transport
(or Brenier maps). In our setting, instead of using complex Monge-Ampère
equations for $(1,1)$ classes as in Theorem 5.7, we apply mass transport (real
Monge-Ampère equations) to convex bodies. Then we reduce the inequality to an
inequality for mixed discriminants, which is easy to prove.

  * Proof.

By the continuity of mixed volumes and taking a limit, it suffices to consider
the case when $K,L,M$ have non-empty interior. We can also assume that the
convex bodies are open. Without loss of generality, we can assume
$\operatorname{vol}(L)=1$.

By [Gro90], for $K,M$, we can take two $C^{2}$ strictly convex functions
$f_{K},f_{M}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ such that

$\operatorname{Image}(\nabla f_{K})=K,\ \ \ \operatorname{Image}(\nabla
f_{M})=M$

where $\nabla$ is the gradient operator. In the following, we denote the
Hessian operator by $\nabla^{2}$.

For $n\times n$ positive matrices $M_{1},...,M_{r}$, the discriminant
$D(M_{1}^{k_{1}},...,M_{r}^{k_{r}})$ is given by the following expansion:

$\det(\sum_{k=1}^{r}t_{k}M_{k})=\sum_{k_{1}+...+k_{r}=n}\frac{n!}{k_{1}!...k_{r}!}D(M_{1}^{k_{1}},...,M_{r}^{k_{r}})t_{1}^{k_{1}}...t_{r}^{k_{r}}.$

Let

$c=\int_{\mathbb{R}^{n}}D((\nabla^{2}f_{K})^{k},(\nabla^{2}f_{M})^{n-k})dx.$

The measure $\rho dx=c^{-1}D((\nabla^{2}f_{K})^{k},(\nabla^{2}f_{M})^{n-k})dx$
is a probability measure on $\mathbb{R}^{n}$. Note that by the argument in
[ADM99] we have

$V(K^{k},M^{n-k})=c.$

By [Bre91] or [McC95], we can take a convex function $F_{L}$ satisfying
$\operatorname{Image}(\nabla F_{L})=L$ and $\det(\nabla^{2}F_{L})=\rho$. As
the argument in [ADM99] again, we have

$\displaystyle V(K^{k},L^{n-k})$
$\displaystyle=\int_{\mathbb{R}^{n}}D((\nabla^{2}f_{K})^{k},(\nabla^{2}F_{L})^{n-k})dx,$
$\displaystyle V(L^{k},M^{n-k})$
$\displaystyle=\int_{\mathbb{R}^{n}}D((\nabla^{2}F_{L})^{k},(\nabla^{2}f_{M})^{n-k})dx.$

As in the proof of Theorem 5.7, by Cauchy-Schwarz inequality we get

$\displaystyle V(K^{k},L^{n-k})V(L^{k},M^{n-k})$
$\displaystyle=\int_{\mathbb{R}^{n}}D((\nabla^{2}f_{K})^{k},(\nabla^{2}F_{L})^{n-k})dx\int_{\mathbb{R}^{n}}D((\nabla^{2}F_{L})^{k},(\nabla^{2}f_{M})^{n-k})dx$
$\displaystyle\geq\left(\int_{\mathbb{R}^{n}}(D((\nabla^{2}f_{K})^{k},(\nabla^{2}F_{L})^{n-k})D((\nabla^{2}F_{L})^{k},(\nabla^{2}f_{M})^{n-k}))^{1/2}dx\right)^{2}.$

Similar to the pointwise inequality (5.6), for the mixed discriminants we have

$\displaystyle
D((\nabla^{2}f_{K})^{k},(\nabla^{2}F_{L})^{n-k})D((\nabla^{2}F_{L})^{k},(\nabla^{2}f_{M})^{n-k})\geq$
$\displaystyle\frac{k!(n-k)!}{n!}\det(\nabla^{2}F_{L})D((\nabla^{2}f_{K})^{k},(\nabla^{2}f_{M})^{n-k}).$

This implies that

$\displaystyle V(K^{k},L^{n-k})V(L^{k},M^{n-k})$
$\displaystyle\geq\left(\int_{\mathbb{R}^{n}}(\frac{k!(n-k)!}{n!}c^{-1}D((\nabla^{2}f_{K})^{k},(\nabla^{2}f_{M})^{n-k})^{2})^{1/2}dx\right)^{2}$
$\displaystyle=\frac{k!(n-k)!}{n!}V(K^{k},M^{n-k})$
$\displaystyle=\frac{k!(n-k)!}{n!}\operatorname{vol}(L)V(K^{k},M^{n-k}),$

where the last equality follows because we assume $\operatorname{vol}(L)=1$.
This finishes the proof of the desired inequality. $\Box$

##  6\. Cone dualities and Minkowski’s existence theorem

In this section, we discuss the convex geometry analogue of the duality of
positive cones in [BDPP13].

###  6.A. The dual cone $\mathcal{C}_{p}^{*}$

For a projective manifold $X$, the movable cone $\operatorname{Mov}_{1}(X)$
has the following characterization:

######  Theorem 6.1 (see [LX16b])

Let $X$ be a smooth projective variety of dimension $n$. Then
$\operatorname{Mov}_{1}(X)$ is the closure of the set $S=\\{\langle
L^{n-1}\rangle|\ L\ \textrm{big and movable class}\\}$. Moreover, the set $S$
is convex.

In summary, by [BDPP13] and [LX16b]:

$\displaystyle\operatorname{Mov}_{1}(X)$ $\displaystyle=\textrm{the closed
cone \emph{genererated} by}\ \\{\langle L_{1}\cdot...\cdot L_{n-1}\rangle|\
L_{i}\ \textrm{big (1,1)-class}\rangle\\}\ \textrm{by
\cite[cite]{[\@@bibref{}{BDPP13, nystrom2016duality}{}{}]}}$
$\displaystyle=\textrm{the closure of \emph{the set}}\ \\{\langle
L^{n-1}\rangle|\ L\ \textrm{big and movable class}\\}\ \textrm{by
\cite[cite]{[\@@bibref{}{lehmann2016positiivty}{}{}]}}.$

Motivated by the correspondence
$\mathcal{C}_{p}\leftrightsquigarrow\operatorname{\overline{Eff}}^{1}(X)$, we
will consider the dual $\mathcal{C}_{p}^{*}$ by analogue with the duality of
cones $\operatorname{\overline{Eff}}^{1}(X)^{*}=\operatorname{Mov}_{1}(X)$.

The dual space of $C^{0}(\mathbb{S}^{n-1})$ is the space of Borel measures on
$\mathbb{S}^{n-1}$. Thus a continuous linear functional on $V$ is a Borel
measure that vanishes on $L(\mathbb{S}^{n-1})$. Indeed, we have

###### Lemma 6.2

A continuous linear functional $\nu$ belongs to $V^{*}$ if and only if $\nu$
is a Borel measure that has the origin as its center of mass, that is,

$\int_{\mathbb{S}^{n-1}}xd\nu(x)=0.$

The Borel measures in the dual of $\mathcal{C}_{p}$ are precisely the set of
nonnegative measures with barycenter $0$. Alexandrov’s version of Minkowski’s
existence theorem yields an alternative description closer to the complex
geometric version above.

To get a feeling for this theorem, let us first recall the original result for
polytopes due to Minkowski: let $u_{1},...,u_{N}$ be pairwise distinct unit
vectors linearly spanning $\mathbb{R}^{n}$, and let $f_{1},...,f_{N}$ be
positive real numbers, then there exists a polytope $P$ (unique up to
translation) having precisely $u_{1},...,u_{N}$ as its outer normal vectors
and having $f_{1},...,f_{N}$ as its corresponded face areas if and only if

$\sum_{i=1}^{N}f_{i}u_{i}=0.$

In particular, if we consider $u_{i}$ as the Dirac measure $\delta_{u_{i}}$ on
the sphere, then $\sum_{i=1}^{N}f_{i}\delta_{u_{i}}$ is a Borel measure on the
sphere with the origin as its center of mass and does not concentrate on any
great subsphere. In the general case, we have the following result due to
Alexandrov [Ale38] (see also [Ale96, Chapter V], [Sch14, Chapter 8]).

######  Theorem 6.3 (Minkowski’s existence theorem)

Assume $\mu$ is a positive Borel measure on $\mathbb{S}^{n-1}$, which is not
concentrated on any great subsphere and satisfies

$\displaystyle\int_{\mathbb{S}^{n-1}}xd\mu(x)=0.$

Then there exists a convex body $K\in\mathfrak{K}^{n}$, unique up to
translation, such that $\mu=S(K,...,K;\cdot)$.

Now we can give the description of $\mathcal{C}_{p}^{*}$, which corresponds
exactly to the structure of the movable cone $\operatorname{Mov}_{1}(X)$,
which follows easily from Minkowski’s existence theorem.

###### Theorem 6.4

The dual cone $\mathcal{C}_{p}^{*}\subseteq V^{*}$ can be characterized as
follows:

$\displaystyle\mathcal{C}_{p}^{*}$ $\displaystyle=\textrm{the closed cone
\emph{generated} by}\
\\{S(K_{1},...,K_{n-1};\cdot)|K_{i}\in\mathfrak{K}^{n}\\}$
$\displaystyle=\textrm{the closure of \emph{the set}}\
\\{S(K^{n-1};\cdot)|K\in\mathfrak{K}^{n}\\}.$

Alternatively, by the embedding $\Phi$, if we consider the mixed volume
$V(K_{1},...,K_{n-1},\cdot)$ as a continuous linear functional on $V$, then

$\displaystyle\mathcal{C}_{p}^{*}$ $\displaystyle=\textrm{the closed cone
\emph{generated} by}\
\\{V(K_{1},...,K_{n-1};\cdot)|K_{i}\in\mathfrak{K}^{n}\\}$
$\displaystyle=\textrm{the closure of \emph{the set}}\
\\{V(K^{n-1};\cdot)|K\in\mathfrak{K}^{n}\\}.$

If we use the notation $\langle h_{K}^{n-1}\rangle:=S(K^{n-1};\cdot)$, then we
have

$\displaystyle\mathcal{C}_{p}^{*}=\textrm{the closure of \emph{the set}}\
\\{\langle P(f)^{n-1}\rangle|f\in\mathcal{C}_{p}\\}.$

  * Proof.

It is obvious that the Borel measures on the right hand side are positive and
are in the dual cone $\mathcal{C}_{p}^{*}$. We only need to verify that any
element in $\mathcal{C}_{p}^{*}$ can be written as a limit of a sequence of
area measures $S(K,...,K;\cdot)$ with $K\in\mathfrak{K}^{n}$.

We claim: let $\nu\in\mathcal{C}_{p}^{*}$, then for any $\epsilon>0$ and
$L\in\mathfrak{K}^{n}$ there exists a convex body $L_{\epsilon}$ such that

$\displaystyle\nu+\epsilon S(L^{n-1};\cdot)=S(L_{\epsilon}^{n-1};\cdot).$

To this end, by Lemma 6.2 we know that $\nu$ has the origin as its centroid:

$\displaystyle\int_{\mathbb{S}^{n-1}}xd\nu(x)=0.$

As $\nu$ is a positive measure, for any $L\in\mathfrak{K}^{n}$ and any
$\epsilon>0$, $\nu+\epsilon S(L^{n-1};\cdot)$ does not concentrate on any
great subsphere. Now the existence of $L_{\epsilon}$ follows immediately from
Minkowski’s existence theorem. $\Box$

By Minkowski’s existence theorem, we have:

###### Proposition 6.5

Assume $K_{1},...,K_{n-1}\in\mathfrak{K}^{n}$ are fixed convex bodies, then
there exists a convex body $K\in\mathfrak{K}^{n}$, unique up to translation,
such that $V(K_{1},...,K_{n-1},\cdot)=V(K^{n-1},\cdot)$.

  * Proof.

We only need to consider the uniqueness, which is an direct corollary of
Brunn-Minkowski theory. $\Box$

###### Remark 6.6

Let $X$ be a smooth projective variety of dimension $n$, and let
$\operatorname{\overline{Eff}}^{1}(X)$ be its pseudo-effective cone of divisor
classes. Thanks to the Newton-Okounkov body construction, if we fix a flag

$\mathcal{F}=(X=H_{0}\supset H_{1}\supset...\supset H_{n-1}\supset
H_{n}=\\{p\\})$

where $H_{i}$ is an irreducible smooth subvariety of $\operatorname{codim}=i$,
then we have a map
$\Delta:\operatorname{\overline{Eff}}^{1}(X)^{\circ}\rightarrow\mathfrak{K}^{n}$.
The map $\Delta$ associates to every big divisor class $L$ a convex body
$\Delta(L)$ with non-empty interior satisfying
$\operatorname{vol}(L)=\operatorname{vol}(\Delta(L))$. In particular, by the
embedding theorem of convex sets, we get a map

$F:\operatorname{\overline{Eff}}^{1}(X)^{\circ}\rightarrow\mathcal{C}.$

Thus the big cone $\operatorname{\overline{Eff}}^{1}(X)^{\circ}$ can be
realized as a finite dimensional convex subcone of $\mathcal{C}$. But $F$ is
only concave. In general, for two big divisor classes $L_{1},L_{2}$, we have
$\Delta(L_{1}+L_{2})\supseteq\Delta(L_{1})+\Delta(L_{2})$, which implies

$F(L_{1}+L_{2})\succeq F(L_{1})+F(L_{2}).$

By relating mixed volumes and positive products of big divisors, it would be
interesting to see whether Minkowski’s existence theorem could provide a new
convex geometry proof of the cone duality
$\operatorname{\overline{Eff}}^{1}(X)^{*}=\operatorname{Mov}_{1}(X)$.

###  6.B. Legendre-Fenchel type transforms

Now that we have the characterization of $\mathcal{C}_{p}^{*}$, we can define
a volume function for all strictly positive continuous functions using a
Legendre-Fenchel type transform as in [LX16a] (see also [Xia15a] for similar
volume characterization of divisors.). Assume that $f\in\mathcal{C}_{p}$ is
represented by a strictly positive continuous function. Define:

$\displaystyle\widetilde{\operatorname{vol}}(f):=\inf_{K\in\mathfrak{K}^{n}}\left(\frac{f\cdot
S(K^{n-1};\cdot)}{\operatorname{vol}(K)^{n-1/n}}\right)^{n}.$

where $f\cdot S(K^{n-1};\cdot)$ is the paring of two dual vector spaces. This
is a homogenous version of the classical polar transform in convex geometry.
Note that by the Brunn-Minkowski inequality, if $f=h_{L}$ for some
$L\in\mathfrak{K}^{n}$ then
$\widetilde{\operatorname{vol}}(h_{L})=\operatorname{vol}(L)$.

We claim that this definition coincides with the definition using the
Alexandrov decomposition described in Section 4. Indeed, let $f=P(f)+N(f)$ be
the decomposition of a strictly positive continuous function. It is clear that

$\widetilde{\operatorname{vol}}(f)\geq\widetilde{\operatorname{vol}}(P(f))=\operatorname{vol}(P(f)).$

Using the orthogonality of the decomposition, we also have

$\widetilde{\operatorname{vol}}(f)\leq\left(\frac{f\cdot\langle
P(f)^{n-1}\rangle}{\operatorname{vol}(P(f))^{n-1/n}}\right)^{n}=\operatorname{vol}(P(f)).$

This establishes the equality.

##  7\. Blaschke addition for (1,1)-classes and its applications

In this section, inspired by Blaschke addition for convex bodies, we introduce
Blaschke addition for (1,1)-classes and give some applications. Throughout we
work with projective varieties.

We start with the convex geometry setting (see [Sch14, Chapter 8]). For convex
bodies $K,L\in\mathfrak{K}^{n}$, Minkowski’s existence theorem yields a convex
body $M\in\mathfrak{K}^{n}$, unique up to translation, such that

$S(M^{n-1};\cdot)=S(K^{n-1};\cdot)+S(L^{n-1};\cdot).$

By the discussions in the previous sections, movable curve classes correspond
to mixed area measures. Let $X$ be a smooth projective variety of dimension
$n$. Assume that $L,M$ are two big (1,1)-classes. As proved in [LX16b], there
exists a unique big and movable (1,1)-class $N$ such that

$\langle L^{n-1}\rangle+\langle M^{n-1}\rangle=\langle N^{n-1}\rangle.$

Now we give the following definition.

######  Definition 7.1 (Blaschke addition)

Let $L,M$ be two big (1,1)-classes. The Blaschke addition of $L,M$, denoted by
$L\\#M$, is defined to be the unique big and movable (1,1)-class satisfying

$\langle(L\\#M)^{n-1}\rangle=\langle L^{n-1}\rangle+\langle M^{n-1}\rangle.$

###### Remark 7.2

By the definition of positive products, we know that $\langle
L^{n-1}\rangle=\langle P(L)^{n-1}\rangle$. Thus Blaschke addition of arbitrary
big classes only depends on their positive parts. For example, for any big
divisor class $L$ we have $L\\#L=2^{\frac{1}{n-1}}P(L)$.

###### Remark 7.3

As pointed out to us by M. Jonsson, the Blaschke addition can also defined on
Riemann-Zariski spaces. More precisely, we can define the Blaschke addition of
two Cartier big $\mathbf{b}$-divisor classes as a Weil movable
$\mathbf{b}$-divisor class. Let $X$ be a smooth projective variety of
dimension $n$ and let $\mathfrak{X}$ be the Riemann-Zariski space containing
all the smooth birational models of $X$. Assume that
$\mathbf{L}=\\{L_{\alpha}\\},\mathbf{M}=\\{M_{\alpha}\\}$ are big Cartier
$\mathbf{b}$-divisor classes on $\mathfrak{X}$. Let
$\varphi_{\alpha\beta}:X_{\alpha}\rightarrow X_{\beta}$ be a birational
morphism. We claim that
$\varphi_{\alpha\beta*}(L_{\alpha}\\#M_{\alpha})=L_{\beta}\\#M_{\beta}$, thus
$\mathbf{L}\\#\mathbf{M}$ is a Weil movable $\mathbf{b}$-divisor class. To
this end, note that by the proof of Proposition 3.3 and the definition of
Cartier divisor class we have

$\displaystyle\varphi_{\alpha\beta}^{*}(\langle L_{\beta}^{n-1}\rangle+\langle
M_{\beta}^{n-1}\rangle)$
$\displaystyle=\langle\varphi_{\alpha\beta}^{*}L_{\beta}^{n-1}\rangle+\langle\varphi_{\alpha\beta}^{*}M_{\beta}^{n-1}\rangle$
$\displaystyle=\langle L_{\alpha}^{n-1}\rangle+\langle
M_{\alpha}^{n-1}\rangle$
$\displaystyle=\langle(L_{\alpha}\\#M_{\alpha})^{n-1}\rangle,$

which implies that
$\langle(L_{\alpha}\\#M_{\alpha})^{n-1}\rangle=\langle\varphi_{\alpha\beta}^{*}(L_{\beta}\\#M_{\beta})^{n-1}\rangle$.
Then we get
$L_{\alpha}\\#M_{\alpha}=\langle\varphi_{\alpha\beta}^{*}(L_{\beta}\\#M_{\beta})\rangle$.
Taking $\varphi_{\alpha\beta*}$ of both sides yields
$\varphi_{\alpha\beta*}(L_{\alpha}\\#M_{\alpha})=\langle
L_{\beta}\\#M_{\beta}\rangle=L_{\beta}\\#M_{\beta}$ (as
$L_{\beta}\\#M_{\beta}$ is movable).

Following [Fir67, Lut86], we can define the mixed divisor class associated to
$(n-1)$ big $(1,1)$-classes.

######  Definition 7.4 (mixed divisor class)

Let $L_{1},...,L_{n-1}$ be big (1,1)-classes on $X$. The mixed (1,1)-class,
denoted by $[L_{1},...,L_{n-1}]$, is defined to be the unique big and movable
(1,1)-class satisfying

$\langle[L_{1},...,L_{n-1}]^{n-1}\rangle=\langle L_{1}\cdot...\cdot
L_{n-1}\rangle.$

Before going on, we first give the following generalized Teissier
proportionality theorem for big and movable $(1,1)$-classes.

###### Theorem 7.5

Let $X$ be a smooth projective variety of dimension $n$. Assume
$L_{1},...,L_{n}$ are big and movable $(1,1)$-classes on $X$, then

$\displaystyle\langle L_{1}\cdot...\cdot
L_{n}\rangle\geq\operatorname{vol}(L_{1})^{1/n}...\operatorname{vol}(L_{n})^{1/n}$

where the equality holds if and only if all the $L_{i}$ are proportional.

  * Proof.

Firstly, by taking suitable Fujita approximations, it is easy to see that the
inequality follows easily from the usual Khovanski-Teissier inequality for nef
classes, hence we only need to characterize the equality situation. Without
loss of generality, we can assume that $\operatorname{vol}(L_{i})=1$ for all
$L_{i}$, then we need to show that

$\displaystyle\langle L_{1}\cdot...\cdot L_{n}\rangle=1$ (7.7)

if and only if $L_{1}=L_{2}=...=L_{n}$. To this end, we claim that the
equality (7.7) implies $\langle L_{1}^{2}\cdot L_{3}...\cdot L_{n}\rangle=1$.
Then by induction, we get that

$\displaystyle\langle L_{1}^{n-1}\cdot L_{n}\rangle=1.$

By the Teissier proportionality theorem proved in [LX16b], we conclude that
$L_{1}=L_{n}$.

The proof of our claim follows from the Hodge-Riemann bilinear relations.
Consider the following quadratic form $Q$ on $H^{1,1}(X,\mathbb{R})$:

$Q(\lambda,\mu):=\lambda\cdot\mu\cdot A_{1}\cdot...\cdot A_{n-2},$

where the $A_{i}$ are Kähler classes. According to the Hodge-Riemann bilinear
relations (see e.g. [DN06, Theorem A and Theorem C]), $Q$ is of signature
$(1,\dim H^{1,1}(X,\mathbb{R})-1)$. For any two nef classes $\alpha,\beta$, by
considering the discriminant of the quadratic polynomial
$p(t):=Q(\alpha+t\beta,\alpha+t\beta)$ and the signature of $Q$, we get

$\displaystyle(\alpha\cdot\beta\cdot A_{1}\cdot...\cdot
A_{n-2})^{2}\geq(\alpha^{2}\cdot A_{1}\cdot...\cdot
A_{n-2})\cdot(\beta^{2}\cdot A_{1}\cdot...\cdot A_{n-2}).$ (7.8)

Since $\alpha,\beta,A_{1},...,A_{n-2}$ are arbitrary, we find by taking
suitable Fujita approximations that

$\displaystyle\langle L_{1}\cdot...\cdot L_{n}\rangle^{2}\geq\langle
L_{1}^{2}\cdot L_{3}...\cdot L_{n}\rangle\cdot\langle L_{2}^{2}\cdot
L_{3}...\cdot L_{n}\rangle.$

Note that, in the above inequality, $\langle L_{1}\cdot...\cdot
L_{n}\rangle=1$ and $\langle L_{j}^{2}\cdot L_{3}...\cdot L_{n}\rangle\geq 1$
for $j=1,2$. Hence we must have equality everywhere, in particular, we must
have $\langle L_{1}^{2}\cdot L_{3}...\cdot L_{n}\rangle=1$. This then finishes
the proof of our claim. $\Box$

###### Remark 7.6

The above proof provides an alternative proof of a related result in [LX16a],
where we applied complex Monge-Ampère equations in big classes and pointwise
Brunn-Minkowski inequalities for positive $(1,1)$-forms.

For Blaschke addition of big (1,1)-classes and mixed (1,1)-classes we obtain
geometric inequalities similar to [Lut86].

###### Proposition 7.7

Let $X$ be a smooth projective variety of dimension $n$. Assume
$L_{1},...,L_{n-1}$ are big and movable $(1,1)$-classes on $X$. Then the
volume of the mixed (1,1)-class $[L_{1},...,L_{n-1}]$ satisfies

$\operatorname{vol}([L_{1},...,L_{n-1}])^{n-1}\geq\operatorname{vol}(L_{1})\cdot...\cdot\operatorname{vol}(L_{n-1})$

where the equality holds if and only if $L_{1},\ldots,L_{n-1}$ are
proportional.

  * Proof.

This is a direct consequence of the Brunn-Minkowski inequality for
(1,1)-classes. Note that we have
$\operatorname{vol}([L_{1},...,L_{n-1}])=\langle[L_{1},...,L_{n-1}]^{n-1}\rangle\cdot[L_{1},...,L_{n-1}]$,
then

$\displaystyle\operatorname{vol}([L_{1},...,L_{n-1}])$ $\displaystyle=\langle
L_{1}\cdot...\cdot L_{n-1}\rangle\cdot[L_{1},...,L_{n-1}]$
$\displaystyle\geq\langle L_{1}\cdot...\cdot
L_{n-1}\cdot[L_{1},...,L_{n-1}]\rangle$
$\displaystyle\geq\operatorname{vol}(L_{1})^{1/n}\cdot...\cdot\operatorname{vol}(L_{n-1})^{1/n}\cdot\operatorname{vol}([L_{1},...,L_{n-1}])^{1/n},$

which implies the desired inequality. The characterization of the equality
follows from Theorem 7.5. $\Box$

We use $[L,M]_{i}$ to denote the mixed (1,1)-class of $\langle L^{n-1-i}\cdot
M^{i}\rangle$. It is clear that $[L,M]_{0}=L$ and $[L,M]_{n-1}=M$.

###### Proposition 7.8

Let $X$ be a smooth projective variety of dimension $n$. Assume that $L,M$ are
big and nef (1,1)-classes. Then the sequence
$\\{\operatorname{vol}([L,M]_{i})\\}_{i=0}^{i=n-1}$ is log-concave, that is,

$\operatorname{vol}([L,M]_{k})^{2}\geq\operatorname{vol}([L,M]_{k-1})\operatorname{vol}([L,M]_{k+1})\
\ \textrm{for any}\ k=1,...,n-2.$

Moreover, the following statements are equivalent:

  1. 1.

$\operatorname{vol}([L,M]_{k})^{2}=\operatorname{vol}([L,M]_{k-1})\operatorname{vol}([L,M]_{k+1})\
\ \textrm{for all}\ k=1,...,n-2$;

  2. 2.

$\operatorname{vol}([L,M]_{n-2})^{n-1}=\operatorname{vol}([L,M]_{0})\operatorname{vol}([L,M]_{n-1})^{n-2}$;

  3. 3.

$L,M$ are proportional.

  * Proof.

By the Hodge-Riemann bilinear relations as in the proof of Theorem 7.5, for
any movable (1,1)-class $N$ we find that

$(L\cdot M\cdot L^{n-2-k}\cdot M^{k-1}\cdot N)^{2}\geq(L^{2}\cdot
L^{n-2-k}\cdot M^{k-1}\cdot N)(M^{2}\cdot L^{n-2-k}\cdot M^{k-1}\cdot N).$

By the definition of mixed class and Brunn-Minkowski inequalities, if we take
$N=[L,M]_{k}$, then we get that

$\displaystyle\operatorname{vol}([L,M]_{k})^{2}$
$\displaystyle\geq(\langle[L,M]_{k-1}^{n-1}\rangle\cdot[L,M]_{k})(\langle[L,M]_{k+1}^{n-1}\rangle\cdot[L,M]_{k})$
$\displaystyle\geq\operatorname{vol}([L,M]_{k-1})^{n-1/n}\operatorname{vol}([L,M]_{k+1})^{n-1/n}\operatorname{vol}([L,M]_{k})^{2/n}.$

This implies the log concavity:
$\operatorname{vol}([L,M]_{k})^{2}\geq\operatorname{vol}([L,M]_{k-1})\operatorname{vol}([L,M]_{k+1})$.

For the equality characterizations, first note that the equivalence of $(2)$
and $(3)$ is a direct consequence of Proposition 7.7. We only need to verify
the equivalence of $(1)$ and $(2)$, but this follows directly from the log
concavity of the sequence $\\{\operatorname{vol}([L,M]_{i})\\}_{i=0}^{i=n-1}$.
$\Box$

By the properties of positive product, Blaschke addition is compatible with
mixed divisor class construction in the following sense.

###### Proposition 7.9

Let $X$ be a projective variety of dimension $n$. Assume that $L,M$ are big
and nef. For any big (1,1)-classes $D_{2},...,D_{n-1}$ we have

$[L+M,D_{2},...,D_{n-1}]=[L,D_{2},...,D_{n-1}]\\#[M,D_{2},...,D_{n-1}]$

where $\\#$ is the Blaschke addition.

  * Proof.

Firstly, by the definition of positive products, we always have

$\displaystyle\langle(L+M)\cdot D_{2}\cdot...\cdot
D_{n-1}\rangle\succeq\langle L\cdot D_{2}\cdot...\cdot D_{n-1}\rangle+\langle
M\cdot D_{2}\cdot...\cdot D_{n-1}\rangle.$

On the other hand, since $L,M$ are nef, we claim that the equality holds: for
any ample class $A$, we have

$\displaystyle\langle(L+M)\cdot D_{2}\cdot...\cdot D_{n-1}\rangle\cdot A$
$\displaystyle=\langle(L+M)\cdot D_{2}\cdot...\cdot D_{n-1}\cdot A\rangle$
$\displaystyle=(L+M)\cdot\langle D_{2}\cdot...\cdot D_{n-1}\cdot A\rangle$
$\displaystyle=\langle L\cdot D_{2}\cdot...\cdot D_{n-1}\rangle\cdot A+\langle
M\cdot D_{2}\cdot...\cdot D_{n-1}\rangle\cdot A.$

By the definition of mixed class, we have

$\displaystyle\langle([L,D_{2},...,D_{n-1}]\\#[M,D_{2},...,D_{n-1}])^{n-1}\rangle$
$\displaystyle=\langle[L,D_{2},...,D_{n-1}]^{n-1}\rangle+\langle[M,D_{2},...,D_{n-1}]^{n-1}\rangle$
$\displaystyle=\langle L\cdot D_{2}\cdot...\cdot D_{n-1}\rangle+\langle M\cdot
D_{2}\cdot...\cdot D_{n-1}\rangle$ $\displaystyle=\langle(L+M)\cdot
D_{2}\cdot...\cdot D_{n-1}\rangle$
$\displaystyle=\langle[L+M,D_{2},...,D_{n-1}]^{n-1}\rangle.$

As a consequence of the Teissier proportionality for big and movable classes
proved in [LX16b], we immediately get the equality
$[L+M,D_{2},...,D_{n-1}]=[L,D_{2},...,D_{n-1}]\\#[M,D_{2},...,D_{n-1}]$.
$\Box$

Similar to the convex geometry setting [Lut86], using Blaschke addition and
mixed divisor classes, we get a somewhat improved Khovanskii-Teissier
inequality for (1,1)-classes.

###### Theorem 7.10

Let $X$ be a smooth projective variety of dimension $n$. Assume that $L,M$ are
big and nef (1,1)-classes, then

$\displaystyle\operatorname{vol}(L+M)^{1/n}$
$\displaystyle\geq\left(\sum_{i=0}^{n-1}\frac{(n-1)!}{i!(n-1-i)!}\operatorname{vol}([L,M]_{i})^{n-1/n}\right)^{1/n-1}$
$\displaystyle\geq\operatorname{vol}(L)^{1/n}+\operatorname{vol}(M)^{1/n}.$

The equality in either inequality holds if and only if $L,M$ are proportional.

  * Proof.

By Proposition 7.9 we have

$[L+M,...,L+M]=\sum_{i=0}^{n-1}\frac{(n-1)!}{i!(n-1-i)!}\cdot[L,M]_{i},$ (7.9)

where the above addition is the Blaschke addition. We claim that

$\operatorname{vol}(L+M)^{n-1/n}\geq\sum_{i=0}^{n-1}\frac{(n-1)!}{i!(n-1-i)!}\operatorname{vol}([L,M]_{i})^{n-1/n}$
(7.10)

with equality if and only if $L,M$ are proportional. To this end, note that

$\displaystyle\operatorname{vol}(L+M)$
$\displaystyle=\langle[L+M,...,L+M]^{n-1}\rangle\cdot(L+M)$
$\displaystyle=\langle(\sum_{i=0}^{n-1}\frac{(n-1)!}{i!(n-1-i)!}\cdot[L,M]_{i})^{n-1}\rangle\cdot(L+M)$
$\displaystyle=\sum_{i=0}^{n-1}\frac{(n-1)!}{i!(n-1-i)!}\langle[L,M]_{i}^{n-1}\rangle\cdot(L+M)$

where the third line follows from the definition of Blaschke addition. Then by
the Brunn-Minkowski inequality, we immediately get the claimed inequality. If
the equality holds, then we must have

$\langle[L,M]_{i}^{n-1}\rangle\cdot(L+M)=\operatorname{vol}([L,M]_{i})^{n-1/n}\operatorname{vol}(L+M)^{1/n}\
\ \textrm{for every}\ i.$

In particular, applying Theorem 7.5 to the cases when $i=0,n-1$, we get that
$L,M$ are proportional.

On the other hand, by Proposition 7.7 we have

$\operatorname{vol}([L,M]_{i})^{n-1}\geq\operatorname{vol}(L)^{n-1-i}\operatorname{vol}(M)^{i}.$

This implies that

$\sum_{i=0}^{n-1}\frac{(n-1)!}{i!(n-1-i)!}\operatorname{vol}([L,M]_{i})^{n-1/n}\geq\left(\operatorname{vol}(L)^{1/n}+\operatorname{vol}(M)^{1/n}\right)^{n-1}.$
(7.11)

For the characterization of the equality, note that the equality implies that,
for every $i$ we have

$\operatorname{vol}([L,M]_{i})^{n-1}=\operatorname{vol}(L)^{n-1-i}\operatorname{vol}(M)^{i}.$

Hence, by Proposition 7.7, we get that $L,M$ must be proportional. $\Box$

###  7.A. Kneser-Süss inequality for divisors

In the convex geometry setting, let $K,L\in\mathfrak{K}^{n}$ be convex bodies,
and let $M$ be the convex body satifying

$S(M^{n-1};\cdot)=S(K^{n-1};\cdot)+S(L^{n-1};\cdot).$

It is proved in [KS32] (see also [Sch14, Chapter 8]) that

$\operatorname{vol}(M)^{n-1/n}\geq\operatorname{vol}(K)^{n-1/n}+\operatorname{vol}(L)^{n-1/n},$

where equality holds if and if $K,L$ are homothetic. This inequality is called
the Kneser-Süss inequality.

Let $X$ be a smooth projective variety of dimension $n$, in [Xia15a] we
defined the following volume type function for movable curve classes:

$\displaystyle\mathfrak{M}(\alpha)=\inf_{B\ \textrm{big
divisor}}\left(\frac{\alpha\cdot
B}{\operatorname{vol}(B)^{1/n}}\right)^{n/n-1},\
\alpha\in\operatorname{Mov}_{1}(X).$

By its definition, we have

$\mathfrak{M}(\alpha_{1}+\alpha_{2})^{n-1/n}\geq\mathfrak{M}(\alpha_{1})^{n-1/n}+\mathfrak{M}(\alpha_{2})^{n-1/n}.$
(7.12)

Later, in [LX16b] (see also [LX16a]), by analyzing “Zariski decomposition
structure” of $\mathfrak{M}$, we have

###### Theorem 7.11

Let $X$ be a smooth projective variety of dimension $n$, and assume that
$\alpha_{i}\in\operatorname{Mov}_{1}(X)$ satisfies
$\mathfrak{M}(\alpha_{i})>0$ ($i=1,2$), then

$\mathfrak{M}(\alpha_{1}+\alpha_{2})^{n-1/n}\geq\mathfrak{M}(\alpha_{1})^{n-1/n}+\mathfrak{M}(\alpha_{2})^{n-1/n}$
(7.13)

with equality if and only if $\alpha_{1},\alpha_{2}$ are proportional.

Equivalently, if we translate the above statement to Blaschke addition for
$(1,1)$-classes, then we get

###### Theorem 7.12

Let $X$ be a smooth projective variety of dimension $n$, and assume that $L,M$
are big movable $(1,1)$-classes, then

$\operatorname{vol}(L\\#M)^{n-1/n}\geq\operatorname{vol}(L)^{n-1/n}+\operatorname{vol}(M)^{n-1/n}$
(7.14)

with equality if and only if $L,M$ are proportional.

##  8\. Projection volumes and the Brascamp-Lieb inequality

There are many geometric inequalities relating the volume of a convex body to
the volumes of its images under linear maps. One of the most well-known is the
Loomis-Whitney inequality [LW49]: let $K$ be a measurable subset of
$\mathbb{R}^{n}$, and let $\pi_{j}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n-1},\
(x_{1},...,x_{n})\mapsto\widehat{x}_{j}:=(x_{1},...,x_{j-1},x_{j+1},...,x_{n})$
be projections, then

$\operatorname{vol}(K)^{n-1}\leq\prod_{j=1}^{n}\operatorname{vol}(\pi_{j}(K)).$

More generally, the Loomis-Whitney inequality finds its natural generalisation
in the Brascamp-Lieb inequality for general linear projections (see e.g.
[BL76], [BCCT08]), but the inequality needs to be corrected by a factor
determined by the projections.

In this section, we formulate and prove an analogue for $(1,1)$ classes.

###  8.A. Projection volumes

Fix a convex body $P\subseteq\mathbb{R}^{n}$ and a unit vector
$v\in\mathbb{R}^{n}$. Then the volume of the orthogonal projection of $P$ onto
$v^{\perp}$ is computed by the mixed volume $n\langle
P^{n-1},\overline{0v}\rangle$, where $\overline{0v}$ is the segment between
$0$ and $v$. The volume of the projection onto the orthogonal of a subspace
$W$ can be computed by a similar procedure.

The analogous construction in complex geometry is as follows. In place of an
orthogonal projection, we fix a dominant rational map $f:X\dashrightarrow Z$
whose resolution has connected fibers. (This is motivated by toric geometry,
where a “direction” in the polytope space identifies a map on a blow-up.)
Recall that [BFJ09] shows that for a big divisor $L$ and a prime divisor $H$
the restricted volume $\operatorname{vol}_{X|H}(L)$ is just the positive
product $\langle L^{n-1}\rangle\cdot H$. Given a big class $L$, the projection
volume onto the fiber is defined to be

$\operatorname{vol}_{f}(L):=\langle\phi^{*}L^{n-d}\rangle\cdot F$

where $d=\dim Z$, $\phi:X^{\prime}\to X$ is a resolution of $f$ and $F$ is a
general fiber of the induced map on $X^{\prime}$. We next need to ask when a
collection $\\{f_{i}:X\dashrightarrow Z_{i}\\}$ is “mutually orthogonal”.
Guided by the toric case, we require that the induced rational map
$X\dashrightarrow\prod_{i}Z_{i}$ is birational.

###  8.B. Loomis-Whitney inequality

Inspired by the Loomis-Whitney inequality for convex bodies, we first prove
the following analogue on projective varieties.

######  Theorem 8.1 (Loomis-Whitney inequality)

Let $X$ be a smooth projective variety of dimension $n$. Suppose that
$\\{f_{i}:X\dashrightarrow C_{i}\\}_{i=1}^{n}$ is a collection of dominant
rational maps onto curves whose resolution has connected fibers, and assume
that the induced map $X\dashrightarrow\prod_{i=1}^{n}C_{i}$ is birational.
Then for any big $(1,1)$ class $L$, we have

$\left(\operatorname{vol}(L)/n!\right)^{n-1}\leq\prod_{i=1}^{n}\left(\operatorname{vol}_{f_{i}}(L)/(n-1)!\right).$

If all the $f_{i}$ are morphisms, then we obtain equality if and only if the
positive part of $L$ is numerically proportional to a sum of fibers of the
$f_{i}$.

Note that the inclusion of the factorials corrects for the discrepancy between
volumes in the complex geometry and in the convex geometry settings.

The first proof of the Loomis-Whitney inequality in the convex geometry
setting was given by [LW49]. The strategy is to approximate a polytope by
hypercubes and to use a counting argument which inducts on dimension. In our
setting, we follow the approach of [SL07], where the authors mainly applied
the method of [Zha99] using mixed volumes.

  * Proof of Theorem 8.1.

After passing to a birational model of $X$ and pulling back $L$, we may
suppose that each rational map $f_{i}$ is a morphism.

Denote the general fiber of $f_{i}$ by $F_{i}$, and assume that
$a_{1},...,a_{n}$ are positive numbers. Note

$\displaystyle\operatorname{vol}(\sum_{i=1}^{n}a_{i}F_{i})=n!\prod_{i=1}^{n}a_{i}.$

By the Khovanski-Teissier inequality for big divisors as in [LX16b], we get

$\displaystyle\langle L^{n-1}\rangle\cdot(\sum_{i=1}^{n}a_{i}F_{i})$
$\displaystyle\geq\operatorname{vol}(L)^{n-1/n}\operatorname{vol}(\sum_{i=1}^{n}a_{i}F_{i})^{1/n}$
$\displaystyle\geq(n!\prod_{i=1}^{n}a_{i})^{1/n}\operatorname{vol}(L)^{n-1/n}$

with equality iff the positive parts of the divisors are numerically
proportional. Setting $a_{i}=c_{i}/\langle L^{n-1}\rangle\cdot F_{i}$ for a
variable $c_{i}$, the above inequality implies that

$\prod_{i=1}^{n}\langle L^{n-1}\rangle\cdot F_{i}\geq
n!\operatorname{vol}(L)^{n-1}\frac{\prod_{i=1}^{n}c_{i}}{(\sum_{i=1}^{n}c_{i})^{n}}.$

Note that $\frac{\prod_{i=1}^{n}c_{i}}{(\sum_{i=1}^{n}c_{i})^{n}}\leq 1/n^{n}$
with equality if and only if all the $c_{i}$ are equal. Letting $c_{i}=1$, we
get

$\prod_{i=1}^{n}\langle L^{n-1}\rangle\cdot
F_{i}/(n-1)!\geq(\operatorname{vol}(L)/n!)^{n-1}.$

Furthermore, equality holds here if and only if equality holds in the
Khovanskii-Teissier inequality above, finishing the proof of our theorem.
$\Box$

###  8.C. Rank-one Brascamp-Lieb inequality

The following statement can be seen as a special case of the Brascamp-Lieb
inequality. In the convex geometry setting, it corresponds to comparing the
volume of a polytope to its projections onto the coordinate axes (instead of
the coordinate hyperplanes).

######  Proposition 8.2 (Rank-one Brascamp-Lieb inequality)

Suppose that $X$ is a smooth projective variety birational to a product of
curves $\prod_{i=1}^{n}C_{i}$. Then for any big class $L$ on $X$ we have

$\operatorname{vol}(L)/n!\leq\prod_{i=1}^{n}\langle L\rangle\cdot T_{i}$

where $T_{i}$ is a general fiber of the rational map
$X\dashrightarrow\prod_{j\neq i}C_{j}$.

  * Proof.

The proof is by induction on the dimension $n$.

By passing to a birational model of $X$ and pulling back $L$, we may suppose
that each rational map $f_{i}$ is a morphism. Note that the quantities
$\langle L\rangle\cdot T_{i}$ do not change upon such pullback, since a very
general fiber $T_{i}$ avoids the restricted base locus of $\langle L\rangle$.
We may also replace $L$ by $\langle L\rangle$ so that we may assume $L$ is big
and movable.

Let $F_{i}$ denote a very general fiber of the map $X\to C_{i}$. Let $s$
denote the largest real number such that $L-sF_{1}$ is pseudo-effective. Note
that $s\leq L\cdot T_{1}$ since $F_{1}\cdot T_{1}=1$ and $(L-sF_{1})\cdot
T_{1}\geq 0$. Then we have

$\operatorname{vol}(L)/n=\int_{0}^{s}\langle(L-tF_{1})^{n-1}\rangle\cdot
F_{1}\,dt\leq(L\cdot T_{1})(\langle L^{n-1}\rangle\cdot F_{1}).$

Clearly $\langle L^{n-1}\rangle\cdot F_{1}$ is at most
$\operatorname{vol}(L|_{F_{1}})$. And by induction, we have

$\operatorname{vol}(L|_{F_{1}})/(n-1)!\leq\prod_{i=2}^{n}\langle
L|_{F_{1}}\rangle\cdot T_{i}.$

Furthermore, for $2\leq i\leq n$ we have $\langle L|_{F_{1}}\rangle\cdot
T_{i}=L\cdot T_{i}$. Indeed, this follows from the fact that $L|_{F_{1}}$ is
movable when $F_{1}$ is very general, so that the positive product on the left
agrees with $L|_{F_{1}}$. We conclude that

$\operatorname{vol}(L)/n\leq(n-1)!\prod_{i=1}^{n}L\cdot T_{i}$

giving the statement. $\Box$

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