# Equality conditions for the fractional Brunn-Minkowski-Lyusternik inequality
in one-dimension

Mark Meyer111Department of Mathematics, Wichita State University, Wichita, KS,
Email<EMAIL_ADDRESS>
###### Abstract

While studying set function properties of Lebesgue measure, Franck Barthe and
Mokshay Madiman proved that Lebesgue measure is fractionally superadditive on
compact sets in $\mathbb{R}^{n}$. In doing this they proved a fractional
generalization of the Brunn-Minkowski-Lyusternik (BML) inequality in dimension
$n=1$. In this article a complete characterization of the equality conditions
for the fractional BML inequality in one-dimension will be given. It will be
shown that aside from some trivial cases, that for a fractional partition
$(\mathcal{G},\beta)$ and non-empty compact sets
$A_{1},\dots,A_{m}\subset\mathbb{R}$ equality holds if and only if for each
$S\in\mathcal{G}$ the set $\sum_{i\in S}A_{i}$ is either an interval with
positive measure or consists of exactly one point.

##  1 Introduction

The Brunn-Minkowski inequality is a well known result of convex geometry. Due
to Hermann Brunn and Hermann Minkowski, the result states for compact, convex
sets $A,B$ in $\mathbb{R}^{n}$ and $\lambda\in[0,1]$ that

$|\lambda
A+(1-\lambda)B|^{\frac{1}{n}}\geq\lambda|A|^{\frac{1}{n}}+(1-\lambda)|B|^{\frac{1}{n}},$

where $|A|$ represents $n$-dimensional Lebesgue measure of a set
$A\subset\mathbb{R}^{n}$. The result was later extended by Lazar Lyusternik
[11] in 1935 to compact sets, and the resulting inequality is called the
Brunn-Minkowski-Lyusternik (BML) inequality.

Theorem 1 [The Brunn-Minkowski-Lyusternik inequality]. Let $A$ and $B$ be non-
empty compact sets in $\mathbb{R}^{n}$. Then

$|A+B|^{\frac{1}{n}}\geq|A|^{\frac{1}{n}}+|B|^{\frac{1}{n}},$

where equality holds if and only if exactly one of the following conditions
holds:
(i) $|A+B|=0$.
(ii) Exactly one of the sets $A,B$ is a point the other has positive measure.
(iii) The sets $A$ and $B$ are full dimensional homothetic convex bodies. (If
$n=1$, then this condition states that $A$ and $B$ are intervals with positive
measure.)

What follows is a summary of some of the motivations for the work in this
paper.

1\. The fractional Young’s inequality and the fractional Brunn-Minkowski
inequality: One result of the BML inequality is that the functional
$A\rightarrow|A|^{\frac{1}{n}}$ is superadditive on compact sets in
$\mathbb{R}^{n}$. This means that if $A_{1},\dots,A_{m}$ are non-empty compact
sets in $\mathbb{R}^{n}$, then

$\left|\sum_{i=1}^{m}A_{i}\right|^{\frac{1}{n}}\geq\sum_{i=1}^{m}|A_{i}|^{\frac{1}{n}}.$

It was conjectured in [3] that there is a fractional generalization of both
Young’s convolution inequality and the Brunn-Minkowski inequality, and it was
shown that the former implies the latter. In particular, it was conjectured
that the above functional is fractionally superadditive. That is, if
$A_{1},\dots,A_{m}$ are compact sets in $\mathbb{R}^{n}$, and if
$(\mathcal{G},\beta)$ is a fractional partition over $\\{1,\dots,m\\}$ (see
Section 2), then

$\left|\sum_{i=1}^{m}A_{i}\right|^{\frac{1}{n}}\geq\sum_{S\in\mathcal{G}}\beta_{S}\left|\sum_{i\in
S}A_{i}\right|^{\frac{1}{n}}.$

We are interested in the generalization of the Brunn-Minkowski inequality. The
interest in this conjecture is that it would give an extensive generalization
of the Brunn-Minkowski inequality when the number of sets being added is
larger than 2. Since fractional superadditivity is an important property in
the study of set functions (see the appendix in [2] for a discussion on this),
this kind of extension for the Brunn-Minkowski inequality would be
interesting. It was immediately verified in [3] that the fractional
superadditivity conjecture is true when the sets are compact and convex. But,
if the sets under consideration are compact, but not necessarily convex, the
fractional BML inequality may not be true. In [7] it was shown that the
fractional BML inequality is not true in deminsion 12 or higher. Later in [6]
it was shown that the conjecture fails in dimension $n\geq 7$. It has been
verified in [2] that the fractional BML conjecture is true in dimension 1.

2\. The monotonicity of the volume deficit: In [7, 8] there are some
developments related to the fractional superadditive BML inequality. The
general idea is to observe the set average of a non-empty compact set
$A\subset\mathbb{R}^{n}$ defined by $A(k)=\frac{1}{k}\sum_{i=1}^{k}A$ and to
observe how close $A(k)$ is to $\text{conv}(A)$. The Shapely-Folkmann-Starr
theorem [17] states that $A(k)$ converges to $\text{conv}(A)$ in Hausdorff
distance. It has also been shown in [5] that the volume deficit
$\Delta(A):=|\text{conv}(A)\backslash A(k)|$ converges to 0 as
$k\rightarrow\infty$. A natural question to ask is whether the volume deficit
is non-increasing in $k$. In [8] the volume deficit is determined to be non-
increasing in dimension 1, but not necessarily for dimension 12 or higher.
Later in [6] it was determined that for star shaped sets the volume deficit is
monotone in $k$ for dimensions 2 and 3, and also for dimension $n\geq 4$ as
long as $k\geq(n-1)(n-2)$. The connection between fractional superadditivity
for the functional $A\rightarrow|A|^{\frac{1}{n}}$ and the monotonicity of the
volume deficit can be seen by observing the fractional superadditive
inequality for the leave-one-out partition (which we know may not be true for
any dimension) and setting each set $A_{i}$ equal to the same set $A$:

$\left|\sum_{i=1}^{m}A\right|^{\frac{1}{n}}\geq\frac{1}{m-1}\sum_{i=1}^{m}\left|\sum_{j=1}^{m-1}A\right|^{\frac{1}{n}}=\frac{m}{m-1}\left|\sum_{i=1}^{m-1}A\right|^{\frac{1}{n}}.$

By rearranging terms and using the n-homogenous property of Lebesgue measure
we find that the above inequality is just $|A(m)|\geq|A(m-1)|$ which then
implies that the volume deficit is non-increasing. It follows that if the
fractional superadditive BML inequality is true, then the volume deficit is
monotone in $k$. On the other hand, if the volume deficit is not non-
increasing for some set $A\subset\mathbb{R}^{n}$, a counter example has been
produced which shows that the fractional superadditive BML inequality for the
leave-one-out partition is not true.

3\. The Shannon-Stam entropy power inequality: There is study of the analogy
between the functional $f_{X}\rightarrow N(X)$, where $X$ is a random vector
with pdf $f_{X}$, $N(X)$ is the power entropy of $X$, and the functional
$A\rightarrow|A|^{\frac{1}{n}}$, where $A$ is compact in $\mathbb{R}^{n}$. A
result in information theory known as the Shannon-Stam entropy power
inequality [15, 16] is analogous to the BML inequality. The Shannon-Stam
entropy power inequality shows that the power entropy $N(X)$ of a random
variable $X$ has a superadditive property: $N(X+Y)\geq N(X)+N(Y)$ if $X$ and
$Y$ are independent random vectors. Later the fractional superadditive
property of the power entropy was determined ([1, 12, 13]).

4\. Supermodularity properties of volume: There is an interest in determining
certain properties of the set function $v(S)=|\sum_{i\in S}A_{i}|$,
$S\subset[m]$. In addition to fractional superadditivity, another property of
interest when studying set functions is supermodularity. The function $v$ is
supermodular if for any $S,T\subset[m]$ it is true that

$v(S\cup T)+v(S\cap T)\geq v(S)+v(T).$

There is a connection between these two properties. If the set function $v$ is
supermodular, then $v$ is also fractionally superadditive. When the sets
$A_{1},\dots,A_{m}$ are compact and convex, the function $v$ is supermodular
[8]. This was determined by proving the inequality

$|A+B+C|+|A|\geq|A+B|+|A+C|,$

which implies that $v$ is supermodular. When the sets $A_{1},\dots,A_{m}$ are
compact and not necessarily convex this inequality was found to be false. As a
result the supermodularity of $v$ could not be verified. But a similar
inequality was verified for dimension 1. If $A,B,C$ are compact sets in
$\mathbb{R}$, then

$|A+B+C|+|\text{conv}(A)|\geq|A+B|+|A+C|.$

In [9] this inequality was verified to be true with certain restrictions when
the dimension is greater than 1.

While it is known that the fractional BML inequality is not always true it
would be interesting to know if at least the volume functional has the
fractional superadditive property when the sets are compact. It would also be
interesting to see that even though the function $v(S)=|\sum_{i\in S}A_{i}|$
is not supermodular on compact sets, that $v$ is fractionally superadditive.
Franck Barthe and Mokshay Madiman in [2] studied volumes of Minkowski sums of
compact sets in $\mathbb{R}^{n}$. They introduced the concept of the
Lyusternik region and as part of their study proved that Lebesgue measure does
have the fractional superadditive property, proving that the fractional BML
inequality in [3] is true for $n=1$ when the sets are compact.

Theorem 2 [Barthe and Madiman]. For an integer $m\geq 3$ let
$(\mathcal{G},\beta)$ be a fractional partition of $[m]$. Then

$|A_{1}+\dots+A_{m}|\geq\sum_{S\in\mathcal{G}}\beta_{S}\left|\sum_{i\in
S}A_{i}\right|$

holds for any non-empty compact subsets $A_{1},\dots,A_{m}$ of
$\mathbb{R}^{n}$.

In [2] it was given as an open question to completely characterize the
conditions for equality when $A_{1},\dots,A_{m}$ are compact subsets of
$\mathbb{R}$. The objective of this article is to provide a solution to this
question.

Some useful references to the Brunn-Minkowski inequality are [4], which
provides a proof of the equality conditions for the BML inequality and [10],
which provides a survey about the Brunn-Minkowski inequality.

Acknowledgements

I would like to thank my PhD advisor Robert Fraser for introducing me to
topics in Convex Geometry and for the assistance he has given me as I worked
on this problem. In particular, he suggested it might be useful to look
carefully at the proof of Theorem 2. In addition, he has taken the time this
summer to read various editions of this article and offered helpful
corrections and advice.

##  2 Notation and definitions

For the following let $A$ represent a compact set in $\mathbb{R}$. If
$a\in\mathbb{R}$, then define ${}_{a\leq}(A):=[a,\infty)\cap A$ and $(A)_{\leq
a}:=(-\infty,a]\cap A$. To denote those $x\in A$ which are the limit of a
decreasing sequence in $A$ we use

$\text{RLP}(A):=\\{x\in A:A\cap(x,x+\varepsilon)\neq\varnothing\text{
}\forall\varepsilon>0\\}.$

To denote the points $x\in A$ which are the limit of an increasing sequence in
$A$ we use

$\text{LLP}(A):=\\{x\in A:A\cap(x-\varepsilon,x)\neq\varnothing\text{
}\forall\varepsilon>0\\}.$

To denote the points $x\in A$ which are surrounded by positive measure in $A$
we use

$\text{PM}(A):=\\{x\in A:|(x-\varepsilon,x+\varepsilon)\cap A|>0\text{
}\forall\varepsilon>0\\}.$

Since the set $\text{PM}(A)$ may not be closed, the set
$\overline{\text{PM}(A)}$ will be used to denote its closure. For
$x\in\text{conv}(A)\backslash\overline{\text{PM}(A)}$ define the closest point
of $\overline{\text{PM}(A)}$ to the right of $x$ by

$x_{R}:=\inf\\{y\in\overline{\text{PM}(A)}:y>x\\},$

and if $\\{y\in\overline{\text{PM}(A)}:y>x\\}=\varnothing$, then we say
$x_{R}$ does not exist. Similarly define the closest point of
$\overline{\text{PM}(A)}$ to the left of $x$ by

$x_{L}:=\sup\\{y\in\overline{\text{PM}(A)}:y<x\\},$

and if $\\{y\in\overline{\text{PM}(A)}:y<x\\}=\varnothing$, then we say
$x_{L}$ does not exist. For the rest of this work, unless stated otherwise, it
will be assumed, because of the translation invariance of Lebesgue measure,
that $\text{min}(A_{i})=0$ and that $\text{max}(A_{i})=a_{i}\geq 0$ for any
compact set $A_{i}\subseteq\mathbb{R}$. To denote the sum $A_{i}+A_{j}$ we
will sometimes use instead $A_{ij}$ and similarly $A_{1}+A_{2}+A_{3}$ will
sometimes be replaced with $A_{123}$. The fractional partition is an important
concept in the following sections.

Definition 1. A fractional partition over $[m]:=\\{1,\dots,m\\}$ is a pair
$(\mathcal{G},\beta)$ where $\mathcal{G}$ is a collection of subsets of $[m]$
and $\beta:2^{[m]}\rightarrow[0,1]$ is a function of weights on $\mathcal{G}$
which satisfies: $\forall i\in[m]$, $\sum_{S\in\mathcal{G}:i\in S}\beta(S)=1$.

To avoid ambiguity three additional requirements will be added to the
definition of a fractional partition. The first requirement is that
$\beta(S)>0$ for each $S\in\mathcal{G}$. We can make this assumption since
removing sets $S\in\mathcal{G}$ with $\beta(S)=0$ has no effect on the
fractional superadditive inequalities in Theorem 2. The second requirement is
that if $(\mathcal{G},\beta)$ is not the trivial partition, then
$[m]\notin\mathcal{G}$. To justify this requirement assume that
$[m]\in\mathcal{G}$ and that $(\mathcal{G},\beta)$ is not the trivial
partition so that $\beta([m])<1$. Then the fractional superadditive inequality
is

$\left|\sum_{[m]}A\right|\geq\beta([m])\left|\sum_{[m]}A\right|+\sum_{S\in\mathcal{G}\backslash[m]}\beta(S)\left|\sum_{S}A\right|.$

This can be simplified to get

$\left|\sum_{[m]}A\right|\geq\sum_{S\in\mathcal{G}\backslash[m]}\frac{\beta(S)}{1-\beta([m])}\left|\sum_{S}A\right|.$

We now observe that for each $S\in\mathcal{G}\backslash[m]$

$\sum_{S\in\mathcal{G}\backslash[m]:i\in
S}\frac{\beta(S)}{1-\beta([m])}=\frac{1-\beta([m])}{1-\beta([m])}=1.$

This shows that every nontrivial fractional partition over $[m]$ corresponds
to a fractional superadditive inequality for which $\beta([m])|\sum_{[m]}A|$
does not belong to the right side of the inequality. It follows that the term
$\beta([m])|\sum_{[m]}A|$ on the right side of the inequality has no influence
over the equality conditions. The third requirement is that the weights
$\beta(S)$ are rational. This assumption allows us to reduce the inequality of
Theorem 2 to the inequality $q|\sum_{[m]}A|\geq\sum_{j=1}^{s}|\sum_{S_{j}}A|$,
where $q\geq 1$ is an integer, the sets $S_{j}$ are an enumeration of the sets
$S\in\mathcal{G}$ possibly with repetition, and each $i\in[m]$ belongs to
exactly $q$ of the sets $S_{j}$. The reason that this assumption can be made
is complicated and the justification is given in Section 6.1

##  3 Initial observations

The objective of this section is to provide an outline of the proof of Theorem
2. We will point out what kind of modifications can be made without causing
the proof to break down. The observations made in this section are crucial in
understanding the proofs of the following results. Recall the assumption that
for each $A_{1},\dots,A_{m}$, $\min(A_{i})=0$ and $\max(A_{i})=a_{i}\geq 0$.
By the remark made at the end of Section 2 the goal is to prove the inequality

$q\left|\sum_{[m]}A\right|\geq\sum_{j=1}^{s}\left|\sum_{S_{j}}A\right|,$

where $q$ is some positive integer, the sets $S_{j}$ are an enumeration of the
sets $S\in\mathcal{G}$ possibly with some repetition, and each $i\in[m]$
belongs to exactly $q$ of the sets $S_{j}$.

To start off suppose that the sets $A_{1}^{\prime},\dots,A_{m}^{\prime}$ are
the sets $A_{1},\dots,A_{m}$ translated to the left by some distances
$\alpha_{i}\leq a_{i}$ such that $0\in A_{i}^{\prime}$, $a_{i}^{\prime}\geq 0$
(this is immediately true since $\alpha_{i}\leq a_{i}$), and for each
$j\in[s]$

$\left|\sum_{S_{j}}A^{\prime}\cap\left(-\infty,0\right]\right|=0.$ (1)

What follows in an outline of the proof of Theorem 2. Some of the steps are
technically difficult and can be seen in full detail in [2].

Outline of the proof. Since each $i\in[m]$ belongs to exactly $q$ of the sets
$S_{j}$, there are indices

$1\leq h_{1}(i)<h_{2}(i)<\dots<h_{q}(i)\leq s$

such that the index $i$ belongs to only the sets $S_{h_{k}(i)}$. With this
notation we observe that since $0\in A_{i}^{\prime}$ and $a_{i}^{\prime}\geq
0$ for each $i\in[m]$, then for $1\leq k\leq q$ the following subset
containments are true.

$\bigcup_{j=1}^{s}\left(\left(\sum_{S_{j}}A^{\prime}+\sum_{h_{k}^{-1}([1,j-1])\backslash
S_{j}}a^{\prime}\right)\cap\left(\sum_{h_{k}^{-1}([1,j-1])}a^{\prime};\sum_{h_{k}^{-1}([1,j])}a^{\prime}\right]\right)\subset\sum_{[m]}A^{\prime}.$
(2)

Since for each $j\in[s]$ the subset containments $h_{k}^{-1}([1,j-1])\subseteq
h_{k}^{-1}([1,j])$ are true, the sets in the union $(2)$ are disjoint. Then by
taking the volumes of the left and right side of $(2)$ we get

$\sum_{j=1}^{s}\left|\left(\sum_{S_{j}}A^{\prime}+\sum_{h_{k}^{-1}([1,j-1])\backslash
S_{j}}a^{\prime}\right)\cap\left(\sum_{h_{k}^{-1}([1,j-1])}a^{\prime};\sum_{h_{k}^{-1}([1,j])}a^{\prime}\right]\right|\leq\left|\sum_{[m]}A^{\prime}\right|.$
(3)

The inequalities $(3)$ hold for each $k\in[q]$. By using the translation
invariance of Lebesgue measure and shifting left by
$\sum_{h_{k}^{-1}([1,j-1])\backslash S_{j}}a^{\prime}$ the inequalities $(3)$
become (full details are in [2])

$\sum_{j=1}^{s}\left|\sum_{S_{j}}A^{\prime}\cap\left(\sum_{h_{k}^{-1}([1,j-1])\cap
S_{j}}a^{\prime};\sum_{h_{k}^{-1}([1,j])\cap
S_{j}}a^{\prime}\right]\right|\leq\left|\sum_{[m]}A^{\prime}\right|.$ (4)

Also, for fixed $j\in[s]$ as $k\in[q]$ varies, the intervals

$\left(\sum_{h_{k}^{-1}([1,j-1])\cap
S_{j}}a^{\prime};\sum_{h_{k}^{-1}([1,j])\cap S_{j}}a^{\prime}\right]$ (5)

are disjoint and their union over $k\in[q]$ is

$\left(0;\sum_{S_{j}}a^{\prime}\right].$

Then in $(4)$ by summing the left and right over $k\in[q]$ we get

$\sum_{j=1}^{s}\left|\sum_{S_{j}}A^{\prime}\cap\left(0;\sum_{S_{j}}a^{\prime}\right]\right|\leq
q\left|\sum_{[m]}A^{\prime}\right|.$

Which by the assumption $(1)$ is

$\sum_{j=1}^{s}\left|\sum_{S_{j}}A^{\prime}\right|\leq
q\left|\sum_{[m]}A^{\prime}\right|.$ (6)

But by translation invariance we have that
$|\sum_{[m]}A^{\prime}|=|\sum_{[m]}A|$ and for each $j\in[s]$ we have that
$|\sum_{S_{j}}A^{\prime}|=|\sum_{S_{j}}A|$. Then $(6)$ can be written as

$\sum_{j=1}^{s}\left|\sum_{S_{j}}A\right|\leq q\left|\sum_{[m]}A\right|.$ (7)

This proves Theorem 2.

Concluding observations. By observing the outline above first observe that
equality in $(7)$ holds if and only if $(6)$ is equality. Also $(6)$ is
equality if and only if $(4)$ is equality for each $k\in[q]$. Moreover, $(4)$
is equality for each $k\in[q]$ if and only if for each $j\in[s]$ and $k\in[q]$

$\begin{split}&\left|\sum_{S_{j}}A^{\prime}\cap\left(\sum_{h_{k}^{-1}([1,j-1])\cap
S_{j}}a^{\prime};\sum_{h_{k}^{-1}([1,j])\cap S_{j}}a^{\prime}\right]\right|\\\
&=\left|\sum_{[m]}A^{\prime}\cap\left(\sum_{h_{k}^{-1}([1,j-1])\cap
S_{j}}a^{\prime};\sum_{h_{k}^{-1}([1,j])\cap
S_{j}}a^{\prime}\right]\right|.\end{split}$ (8)

When $j=k=1$ note that $h_{1}^{-1}(\\{1\\})=S_{1}$ and that $(8)$ becomes

$\left|\sum_{S_{1}}A^{\prime}\cap\left(0,\sum_{S_{1}}a^{\prime}\right]\right|=\left|\sum_{[m]}A^{\prime}\cap\left(0,\sum_{S_{1}}a^{\prime}\right]\right|.$

Then if we show that

$\left|\sum_{S_{1}}A^{\prime}\cap\left(0,\sum_{S_{1}}a^{\prime}\right]\right|<\left|\sum_{[m]}A^{\prime}\cap\left(0,\sum_{S_{1}}a^{\prime}\right]\right|$

we have shown that $(8)$ is a strict inequality when $j=k=1$ and therefore
that the inequality $(7)$ is a strict inequality. This observation will be
used frequently throughout the article.

##  4 Equality conditions when all sets have positive measure

In this section we will observe the equality conditions for when each of the
sets $\sum_{S}A$ has positive measure. There are some technical lemmas and the
result is established in Theorem 3. For this section and the remaining
sections we recall the assumptions made in the introduction that the weights
in the fractional partitions are non-zero rational numbers and that a non-
trivial fractional partition of $[m]$ does not contain the set $[m]$.

Lemma 1. Let $A_{1},\dots,A_{m}$ be non-empty compact sets in $\mathbb{R}$ and
let $(\mathcal{G},\beta)$ be a fractional partition of $[m]$. Then

$\left|\text{conv}\left(\sum_{[m]}A\right)\right|=\sum_{S\in\mathcal{G}}\beta_{S}\left|\text{conv}\left(\sum_{S}A\right)\right|.$

Proof: To prove this lemma we will make use of the fact that for sets $A$, $B$
in $\mathbb{R}$, $\text{conv}(A+B)=\text{conv}(A)+\text{conv}(B)$. Observe
that

$\begin{split}\left|\text{conv}\left(\sum_{[m]}A\right)\right|&=\left|\sum_{i=1}^{m}\text{conv}(A_{i})\right|\\\
&=\left|\sum_{i=1}^{m}\left[\sum_{S\in\mathcal{G}:i\in
S}\beta_{S}\cdot\text{conv}(A_{i})\right]\right|\\\
&=\left|\sum_{i=1}^{m}\sum_{S\in\mathcal{G}}\beta_{S}\textbf{I}_{i}(S)\text{conv}(A_{i})\right|\\\
&=\left|\sum_{S\in\mathcal{G}}\beta_{S}\sum_{i\in
S}\text{conv}(A_{i})\right|\\\
&=\sum_{S\in\mathcal{G}}\beta_{S}\left|\text{conv}\left(\sum_{S}A\right)\right|.\end{split}$

This completes proof of the lemma.

Lemma 2. Let $A_{1},\dots,A_{m}$ be non-empty compact sets in $\mathbb{R}$ and
let $(\mathcal{G},\beta)$ be a fractional partition over $[m]$. If each of the
sets $\sum_{S}A$ is an interval, then the set $\sum_{[m]}A$ is also an
interval and equality is true:

$\left|\sum_{[m]}A\right|=\sum_{S\in\mathcal{G}}\beta_{S}\left|\sum_{S}A\right|.$

Proof: If each $\sum_{S}A$ is an interval, then by Lemma 1

$\begin{split}\left|\sum_{[m]}A\right|&\leq\left|\text{conv}\left(\sum_{[m]}A\right)\right|\\\
&=\sum_{S\in\mathcal{G}}\beta_{S}\left|\text{conv}\left(\sum_{S}A\right)\right|\\\
&=\sum_{S\in\mathcal{G}}\beta_{S}\left|\sum_{S}A\right|\\\
&\leq\left|\sum_{[m]}A\right|.\end{split}$

It follows that all of the $\leq$ must be $=$ which proves that equality is
true. Also we see that

$\left|\sum_{[m]}A\right|=\left|\text{conv}\left(\sum_{[m]}A\right)\right|,$

which by compactness implies that
$\sum_{[m]}A=\text{conv}\left(\sum_{[m]}A\right)$. This proves that the set
$\sum_{[m]}A$ must be an interval.

Remark 1. In Lemma 2 the method of showing that $\sum_{[m]}A$ must be an
interval works only because of the fact that for compact intervals
$I_{1},I_{2}\subset\mathbb{R}$ it is true that
$|I_{1}+I_{2}|=|I_{1}|+|I_{2}|$. In higher dimension if $A,B$ are convex and
compact, then it is not always true that $|A+B|=|A|+|B|$. To prove the
analogue of Lemma 2 in dimension $n\geq 2$, that $\sum_{[m]}A$ is convex if
the sets $\sum_{S}A$ are convex for each $S\in\mathcal{G}$, there is a useful
inequality involving the Schneider non-convexity index which can be used. See
Section 6.2 for more details.

Lemma 3. Let $m\geq 3$ be an integer and let $(\mathcal{G},\beta)$ be a
fractional partition on $[m]$ that is not the trivial partition. For
$k\in(1,m)$ define a set

$\mathcal{G^{*}}:=\\{S\cap[k]\notin\\{\varnothing,[k]\\}:S\in\mathcal{G}\\}.$

Define the number $\gamma$ by

$\gamma:=\sum_{S\in\mathcal{G}:S\cap[k]=[k]}\beta(S).$

Also assume that $\gamma<1$. Now define a function
$\beta^{*}:\mathcal{G}^{*}\rightarrow[0,1]$ by

$\beta^{*}(T):=\frac{1}{1-\gamma}\sum_{S\in\mathcal{G}:S\cap[k]=T}\beta(S).$

Then the pair $(\mathcal{G}^{*},\beta^{*})$ is a non-trivial fractional
partition of $[k]$.

Proof: Let $i\in[k]$. Then

$\begin{split}\sum_{T\in\mathcal{G}^{*}:i\in
T}\beta^{*}(T)&=\frac{1}{1-\gamma}\sum_{T\in\mathcal{G}^{*}:i\in
T}\left[\sum_{S\in\mathcal{G}:S\cap[k]=T}\beta(S)\right]\\\
&=\frac{1}{1-\gamma}\sum_{T\in\mathcal{G}^{*}}\left[\sum_{S\in\mathcal{G}:S\cap[k]=T}\beta(S)\textbf{I}_{i}(T)\right]\\\
&=\frac{1}{1-\gamma}\sum_{T\in\mathcal{G}^{*}}\left[\sum_{S\in\mathcal{G}:S\cap[k]=T}\beta(S)\textbf{I}_{i}(S)\right]\\\
&=\frac{1}{1-\gamma}\sum_{S\in\mathcal{G}:S\cap[k]\notin\\{\varnothing,[k]\\}}\beta(S)\textbf{I}_{i}(S)\\\
&=\frac{1}{1-\gamma}\left(1-\sum_{S\in\mathcal{G}:S\cap[k]=[k]}\beta(S)\textbf{I}_{i}(S)\right)=1.\end{split}$

This verifies that the pair $(\mathcal{G}^{*},\beta^{*})$ is a fractional
partition of $[k]$. The fact that this partition is not the trivial partition
follows from the fact that each set $T\in\mathcal{G}^{*}$ is a proper subset
of $[k]$ and so there must be at least two sets in $\mathcal{G}^{*}$ in order
to cover all the integers in $[k]$.

Remark 2. Suppose that $m\geq 3$ is an integer, that $(\mathcal{G},\beta)$ is
a non-trivial fractional partition of $[m]$, and that $A_{1},\dots,A_{m}$ are
non-empty compact sets in $\mathbb{R}$. The corresponding fractional
superadditive inequality is given by

$\left|\sum_{[m]}A\right|\geq\sum_{S\in\mathcal{G}}\beta(S)\left|\sum_{S}A\right|.$

Suppose that for some $k\in(1,m)$ exactly $k$ of the sets have
$\text{card}(A_{i})\geq 2$ and the remaining $m-k$ sets have
$\text{card}(A_{i})=1$. We will suppose that the sets $A_{1},\dots,A_{k}$ are
the sets with two or more points each, and the sets $A_{k+1},\dots,A_{m}$ are
the sets which are points. Using the notation of Lemma 3, we can rewrite this
fractional superadditive inequality as

$\left|\sum_{[k]}A\right|\geq\gamma\left|\sum_{[k]}A\right|+\sum_{S\in\mathcal{G}:S\cap[k]\notin\\{\varnothing,[k]\\}}\beta(S)\left|\sum_{S\cap[k]}A\right|.$

If $\gamma=1$, then the inequality is $|\sum_{[k]}A|\geq|\sum_{[k]}A|$. This
is the fractional superadditive inequality for the trivial partition on $[k]$:
$\mathcal{G}^{*}=\\{[k]\\}$, $\beta^{*}([k])=1$. If $\gamma<1$ we find that
the inequality becomes

$\left|\sum_{[k]}A\right|\geq\sum_{S\in\mathcal{G}:S\cap[k]\notin\\{\varnothing,[k]\\}}\frac{\beta(S)}{1-\gamma}\left|\sum_{S\cap[k]}A\right|.$

By the definitions of $\mathcal{G}^{*}$ and $\beta^{*}$ this is

$\left|\sum_{[k]}A\right|\geq\sum_{T\in\mathcal{G}^{*}}\beta^{*}(T)\left|\sum_{T}A\right|.$

The result is that by using the translation invariance of Lebesgue measure, we
have rewritten the original inequality so that each of the sets on the right
hand side $\sum_{T}A$ contain at least two points. For future reference, the
new fractional partition $(\mathcal{G}^{*},\beta^{*})$ will be called the
translated fractional partition on $[k]$ sets. That is, the new fractional
partition that is obtained by removing all of the sets $A_{i}$ which contain
only one point.

Theorem 3. Let $m\geq 3$ be an integer, let $(\mathcal{G},\beta)$ be a
fractional partition on $[m]$ that is not the trivial partition
$\beta([m])=1$, and let $A_{1},\dots,A_{m}$ be non-empty compact sets in
$\mathbb{R}$ such that $|\sum_{S}A|>0$ for any $S\in\mathcal{G}$. Then

$\left|\sum_{[m]}A\right|=\sum_{S\in\mathcal{G}}\beta_{S}\left|\sum_{S}A\right|$

if and only if for any $S\in\mathcal{G}$, $\sum_{S}A$ is an interval.

Proof: Suppose for each $S\in\mathcal{G}$ that $\sum_{S}A$ is an interval. By
Lemma 2 equality must be true.

Now assume that equality is true and assume by induction that Theorem 3 is
true for $k<m$ sets. First, consider the case for when $k$ of the sets have
$\text{card}(A_{i})\geq 2$ and the remaining $m-k$ sets have
$\text{card}(A_{i})=1$. Rearranging the sets if needed, we may assume that
$A_{1},\dots,A_{k}$ have $\text{card}(A_{i})\geq 2$ and that
$A_{k+1},\dots,A_{m}$ have $\text{card}(A_{i})=1$. Using the notation of Lemma
3 we have that

$\gamma:=\sum_{S\in\mathcal{G}:S\cap[k]=[k]}\beta(S).$

We must have that $\gamma<1$. Assume for contradiction that $\gamma=1$. Since
$(\mathcal{G},\beta)$ is not the trivial partition there exists an integer
$t\in[k]$ and a set $S^{(t)}\in\mathcal{G}$ such that $t\notin S^{(t)}$ (If
$t$ belongs to each $S\in\mathcal{G}$, then $\sum_{S\in\mathcal{G}:t\in
S}\beta(S)>1$). The set $S^{(t)}\cap[k]$ is non-empty. Otherwise the set
$\sum_{S^{(t)}}A$ would contain only point sets and it would follow that
$|\sum_{S^{(t)}}A|=0$ which is a contradiction. Choose $j\in S^{(t)}\cap[k]$.
Then since $S^{(t)}\cap[k]$ is a proper subset of $[k]$ we have

$\sum_{S\in\mathcal{G}:j\in S}\beta(S)\geq\beta(S^{(t)})+\gamma>1.$

This is a contradiction, so we must have $\gamma<1$. Now using Lemma 3 and
Remark 2 the equality becomes

$\left|\sum_{[k]}A\right|=\sum_{T\in\mathcal{G}^{*}}\beta^{*}(T)\left|\sum_{T}A\right|,$

where the pair $(\mathcal{G}^{*},\beta^{*})$ is a non-trivial fractional
partition of $[k]$. By induction each of the sets $\sum_{T}A$ is an interval
of positive measure. By Lemma 2 the set $\sum_{[k]}A$ is an interval of
positive measure. By translation each of the sets $\sum_{S}A$ must be
intervals of positive measure. This proves Theorem 3 for when some of the sets
have $\text{card}(A_{i})=1$.

Next we will consider the case where $\text{card}(A_{i})\geq 2$ for all
$i\in[m]$, at least one of the sets $\sum_{S}A$ is an interval and at least
one of them is not an interval. Using the assumption that the fractional
partition has rational weights the equality can be written as

$q\left|\sum_{[m]}A\right|=\sum_{j=1}^{s}\left|\sum_{S_{j}}A\right|,$

where $q$ is some positive integer and the sets $S_{j}$ are an enumeration of
the sets $S\in\mathcal{G}$ possibly with some repetition. For the following
claim we will define $T_{j}:=\sum_{S_{j}}A$.

Claim. There exists a set $T_{j}$ which is not an interval and a set $T_{k}$
which is an interval such that one of the sets $A_{i}$ in the sum $T_{k}$ is
not one of the sets in the sum $T_{j}$.

Proof of the claim: By assumption there exists a set $T_{j}$ which is not an
interval. Assume that each set $T_{k}$ which is an interval is a partial sum
of the set $T_{j}$ (or else we are done). Since $T_{j}$ does not have all $m$
sets in its sum, there exists a set $A_{t}$ which is not in the sum $T_{j}$.
There must exist a set $T_{j}^{\prime}$ which contains $A_{t}$ in its sum. Let
$T_{k}$ be one of the sets which is an interval. This set contains some set
$A_{i}$ in its sum. If $A_{i}$ is contained in all those sets $T_{j}^{\prime}$
which have $A_{t}$ in their sum, then the weights for $i$ would sum to larger
than $1$. So there must exist a set $T_{j}^{\prime}$ which contains $A_{t}$ in
its sum (so it cannot be an interval) which does not contain $A_{i}$ in its
sum. This proves the claim.

Using the claim let $T_{j}$ be one of the sets which is not an interval and
let $T_{k}$ be one of the sets that is an interval which has the set $A_{i}$
in its sum such that $A_{i}$ is not in the sum of $T_{j}$. By rearranging the
sets $S_{j}$ we can assume that $T_{j}=T_{1}$. If the set
$(\sum_{[m]}A)_{\leq\sum_{S_{1}}a}$ is an interval, then

$\left|(\sum_{[m]}A)_{\leq\sum_{S_{1}}a}\right|=\left|\text{conv}\left(\sum_{S_{1}}A\right)\right|>\left|\sum_{S_{1}}A\right|$

which is a contradiction of $(8)$ with $k=1$ where the sets $A_{i}$ have not
been shifted. If the set $(\sum_{[m]}A)_{\leq\sum_{S_{1}}a}$ is not an
interval, then

$\begin{split}\left|(\sum_{[m]}A)_{\leq\sum_{S_{1}}a}\right|&=\left|\left(\text{conv}(\sum_{S_{k}}A)+\sum_{[m]\backslash
S_{k}}A\right)_{\leq\sum_{S_{1}}a}\right|\\\
&\geq\left|\left(\text{conv}(\sum_{S_{k}}A)+\sum_{([m]\backslash S_{k})\cap
S_{1}}A\right)_{\leq\sum_{S_{1}}a}\right|\\\
&\geq\left|\left(\text{conv}(A_{i})+\sum_{S_{1}}A\right)_{\leq\sum_{S_{1}}a}\right|\\\
&>\left|\sum_{S_{1}}A\right|.\end{split}$

The last inequality is strict because $\text{card}(A_{i})\geq 2$ so
$\text{conv}(A_{i})$ is an interval with positive measure which is then added
to a set with open intervals in its complement. This again proves a
contradiction to $(8)$ with $k=1$, where the sets $A_{i}$ have not been
shifted left.

Finally, we will assume that all the sets $T_{j}:=\sum_{S_{j}}A$ are not
intervals. For each $j\in[s]$ define

$\varepsilon_{j}:=\sup\\{\varepsilon>0:|T_{j}\cap[0,\varepsilon)|=0\\}.$

Suppose first for each $j\in[s]$ that $\varepsilon_{j}>0$. Choose
$j_{0}\in[s]$ such that $\varepsilon_{j_{0}}\leq\varepsilon_{j}$ for each
$j\in[s]$. By compactness $\varepsilon_{j_{0}}\in T_{j_{0}}$. Then
$\varepsilon_{j_{0}}=\sum_{i\in S_{j_{0}}}\alpha_{i}$, where $\alpha_{i}\in
A_{i}$ for each $i\in S_{j_{0}}$. Choose some $i_{0}\in S_{j_{0}}$ such that
$\alpha_{i_{0}}>0$. Now shift the set $A_{i_{0}}$ to the left by
$\alpha_{i_{0}}$ to achieve a new set
$A_{i_{0}}^{\prime}:=A_{i_{0}}-\alpha_{i_{0}}$. Clearly after shifting left
$0\in A_{i_{0}}^{\prime}$ and $|T_{j}^{\prime}\cap(-\infty,0]|=0$ for each
$j\in[s]$. There exists a set $T_{j}$ such that $i_{0}\notin S_{j}$. By
rearranging we may assume that this set is $T_{1}$. First note that
$T_{1}+A_{i_{0}}^{\prime}\supseteq T_{1}$ because $0\in A_{i_{0}}^{\prime}$.
Also because $A_{i_{0}}$ has some points to the left of $0$,
$|(T_{1}+A_{i_{0}}^{\prime})\cap(0,\varepsilon_{1}]|>0$. Then

$\begin{split}\left|(\sum_{[m]}A^{\prime})_{\leq\sum_{S_{1}}a}\right|&\geq\left|(T_{1}+A_{i_{0}}^{\prime})_{\leq\sum_{S_{1}}a}\right|\\\
&\geq|(T_{1}+A_{i_{0}}^{\prime})\cap(0,\varepsilon_{1}]|+|T_{1}|\\\
&>|T_{1}|.\end{split}$

This is a contradiction to $(8)$. Next suppose for at least one $j\in[s]$ that
$\varepsilon_{j}=0$ and for at least one $j\in[s]$ that $\varepsilon_{j}>0$.
Choose $j_{1},j_{2}\in[s]$ such that $\varepsilon_{j_{1}}=0$ and
$\varepsilon_{j_{2}}>0$. By rearranging we may assume that $j_{2}=1$. By
choice of $j_{1}$ it follows that $|T_{j_{1}}\cap[0,\varepsilon_{1})|>0$. Then
it follows that

$\left|\sum_{[m]}A\cap[0,\varepsilon_{1})\right|>0.$

Therefore

$\left|(\sum_{[m]}A)_{\leq\sum_{S_{1}}a}\right|\geq\left|\sum_{[m]}A\cap[0,\varepsilon_{1})\right|+|T_{1}|>|T_{1}|.$

This is a contradiction to $(8)$. Finally suppose that $\varepsilon_{j}=0$ for
each $j\in[s]$. Then there exists $i\in[m]$ such that $0\in\text{RLP}(A_{i})$.
Also there is $T_{j}$ such that $i\notin S_{j}$. By rearranging we may assume
that $T_{j}=T_{1}$. Now, there exists
$x\in\text{conv}(T_{1})\backslash\overline{\text{PM}(T_{1})}$ such that
$x_{L}$ exists. Otherwise $T_{1}$ would be an interval. Choose such a number
$x$. Then adding $A_{i}$ to $T_{1}$ causes a small piece of
$\overline{\text{PM}(T_{1})}$ to shift into the interval $(x_{L},x)$. That is,
$|(A_{i}+T_{1})\cap(x_{L},x)|>0$. Therefore, since $0\in A_{i}$ we have

$\begin{split}\left|(\sum_{[m]}A)_{\leq\sum_{S_{1}}a}\right|&\geq|(A_{i}+T_{1})_{\leq\sum_{S_{1}}a}|\\\
&\geq|(A_{i}+T_{1})\cap(x_{L},x)|+|T_{1}|\\\ &>|T_{1}|.\end{split}$

Again, this is a contradiction to $(8)$. This completes the proof.

##  5 The complete characterization of the equality conditions

By using Theorem 3 and the fact stated in Lemma 4 below that the sets
$\sum_{S}A$ with zero measure must be points, we can find the equality
conditions for a general fractional partition on $m$ sets.

Lemma 4. Let $A_{1},\dots,A_{m}$ be non-empty compact sets in $\mathbb{R}$ and
let $(\mathcal{G},\beta)$ be a non-trivial fractional partition of $[m]$.
Suppose that $|\sum_{[m]}A|>0$ and that

$\left|\sum_{[m]}A\right|=\sum_{S\in\mathcal{G}}\beta(S)\left|\sum_{S}A\right|.$

If $\left|\sum_{S}A\right|=0$, then $\text{card}\left(\sum_{S}A\right)=1$.

Proof: First recall that the equality can be written as

$q\left|\sum_{[m]}A\right|=\sum_{j=1}^{s}\left|\sum_{S_{j}}A\right|,$

where $q$ is a positive integer and the $S_{j}$ are an enumeration of the sets
$S\in\mathcal{G}$ possibly with repetition. Since $(\mathcal{G},\beta)$ is a
non-trivial fractional partition there exist at least two distinct sets
$S_{j}$. Assume for contradiction that there exists a set $S_{j}$ with
$\text{card}\left(\sum_{S_{j}}A\right)>1$ and $|\sum_{S_{j}}A|=0$. By
rearranging if needed, we may assume that this set is $S_{1}$. Now let
$S_{1}^{\prime},\dots,S_{k}^{\prime}$ denote those sets for which
$|\sum_{S_{j}^{\prime}}A|>0$. We first assume that at least one of those sets
satisfies

$\left|\sum_{S_{j}^{\prime}}A\cap(0,\varepsilon]\right|>0$

for all $\varepsilon>0$. Since equality is true we must have (see Section 3
equation $(8)$ with $j=k=1$)

$\left|\sum_{[m]}A\cap\left(0,\sum_{S_{1}}a\right]\right|=\left|\sum_{S_{1}}A\cap\left(0,\sum_{S_{1}}a\right]\right|=0.$

But this is a contradiction because

$\left|\sum_{[m]}A\cap\left(0,\sum_{S_{1}}a\right]\right|\geq\left|\sum_{S_{j}^{\prime}}A\cap\left(0,\sum_{S_{1}}a\right]\right|>0.$

It was used that $\sum_{S_{1}}a>0$ which has to be true since
$\text{card}\left(\sum_{S_{1}}A\right)>1$. Finally, assume that for each
$j\in[k]$ that there exists $\varepsilon>0$ such that

$\left|\sum_{S_{j}^{\prime}}A\cap(0,\varepsilon]\right|=0.$

For each $j\in[k]$ set

$\varepsilon_{j}:=\sup\left\\{\varepsilon>0:\left|\sum_{S_{j}^{\prime}}A\cap(0,\varepsilon]\right|=0\right\\}.$

Then choose $j_{0}$ such that $\varepsilon_{j_{0}}\leq\varepsilon_{j}$ for
each $j\in[k]$. By compactness,
$\varepsilon_{j_{0}}\in\sum_{S_{j_{0}}^{\prime}}A$. Then there exist numbers
$\alpha_{i}\in A_{i}$ for each $i\in S_{j_{0}}^{\prime}$ such that
$\varepsilon_{j_{0}}=\sum_{i\in S_{j_{0}}^{\prime}}\alpha_{i}$. If $i\in
S_{j_{0}}^{\prime}\backslash S_{1}$, then replace the set $A_{i}$ with
$A_{i}^{\prime}:=A_{i}-\alpha_{i}$. With these replaced sets we still have for
each $j\in[m]$ that $0\in A_{i}^{\prime}$ and $a_{i}^{\prime}\geq 0$. We also
have for each $j\in[s]$ that $|(\sum_{S_{j}}A^{\prime})_{\leq 0}|=0$. After
this replacement of sets let $\varepsilon_{j_{0}}^{\prime}$ be defined the
same as the $\varepsilon_{j}$ above, but with the replaced sets
$A_{i}^{\prime}$. In fact,

$\varepsilon_{j_{0}}^{\prime}=\varepsilon_{j_{0}}-\sum_{i\in
S_{j_{0}}^{\prime}\backslash S_{1}}\alpha_{i}=\sum_{i\in
S_{j_{0}}^{\prime}\cap S_{1}}\alpha_{i}\leq\sum_{S_{1}}a.$

If $\varepsilon_{j_{0}}^{\prime}<\sum_{S_{1}}a$, then with the replaced sets,

$\left|\sum_{[m]}A^{\prime}\cap\left(0,\sum_{S_{1}}a\right]\right|\geq\left|\sum_{S_{j_{0}}^{\prime}}A^{\prime}\cap\left(0,\sum_{S_{1}}a\right]\right|>0$

which is a contradiction since $|\sum_{S_{1}}A|=0$. Note that there was no
need to replace $\sum_{S_{1}}a$ by $\sum_{S_{1}}a^{\prime}$ since none of the
sets $A_{i}$ with $i\in S_{1}$ were modified. Now suppose that
$\varepsilon_{j_{0}}^{\prime}=\sum_{S_{1}}a$. For at least one $i\in
S_{j_{0}}^{\prime}\backslash S_{1}$, $\alpha_{i}\in\text{RLP}(A_{i})$. Choose
any one such $i$, then instead replace that set $A_{i}$ by
$A_{i}^{\prime\prime}:=A_{i}-(\alpha_{i}+\varepsilon)$, where
$\alpha_{i}+\varepsilon\in A_{i}$ and $\varepsilon\in(0,\sum_{S_{1}}a)$. With
this replacement notice that we still have that
$|(\sum_{S_{j}}A^{\prime\prime})_{\leq 0}|=0$ for each $j\in[s]$ and that

$0<\varepsilon_{j_{0}}^{\prime\prime}=\varepsilon_{j_{0}}^{\prime}-\varepsilon<\sum_{S_{1}}a$

and we may repeat the technique used above to find a contradiction.

Theorem 4. Let $A_{1},\dots,A_{m}$ be non-empty compact sets in $\mathbb{R}$
and let $(\mathcal{G},\beta)$ be a non-trivial fractional partition of $[m]$.
Then

$\left|\sum_{[m]}A\right|=\sum_{S\in\mathcal{G}}\beta_{S}\left|\sum_{S}A\right|$

if and only if exactly one of the conditions is true:
(A) $\left|\sum_{[m]}A\right|=0$
(B) $\left|\sum_{[m]}A\right|>0$, some $k\in[1,m)$ of the sets have
$\text{card}(A_{i})\geq 2$, the remaining $m-k$ sets have
$\text{card}(A_{i})=1$, and the translated fractional partition (see remark 2)
is the trivial partition on $[k]$.
(C) $\left|\sum_{[m]}A\right|>0$, some $k\in(1,m]$ of the sets have
$\text{card}(A_{i})\geq 2$, the remaining $m-k$ sets have
$\text{card}(A_{i})=1$, each set $\sum_{S}A$ with positive measure is an
interval, each set $\sum_{S}A$ with measure zero is a point, and the
translated fractional partition is non-trivial.

Proof: Assume that $|\sum_{[m]}A|>0$ and that for some $k\in[1,m)$ exactly $k$
of the sets have $\text{card}(A_{i})\geq 2$ and the remaining $m-k$ sets have
$\text{card}(A_{i})=1$. We will assume that the sets with
$\text{card}(A_{i})\geq 2$ are $A_{1},\dots,A_{k}$ and the sets with
$\text{card}(A_{i})=1$ are $A_{k+1},\dots,A_{m}$. By Remark 2, the equality
can be written as

$\left|\sum_{[k]}A\right|=\sum_{T\in\mathcal{G}^{*}}\beta^{*}(T)\left|\sum_{T}A\right|.$

Using the notation of Lemma 3, if $\gamma=1$, then the partition
$(\mathcal{G}^{*},\beta^{*})$ is the trivial partition
$\mathcal{G}^{*}=\\{[k]\\}$, $\beta^{*}([k])=1$. This verifies condition
$(B)$. If $\gamma<1$, then by Lemma 3, the partition
$(\mathcal{G}^{*},\beta^{*})$ is a non-trivial fractional partition of $[k]$
and each of the sets $\sum_{T}A$ contain at least two points. By Lemma 4 each
of the sets $\sum_{T}A$ has positive measure. Then by Theorem 3 each of the
sets $\sum_{T}A$ must be an interval of positive measure. By Lemma 2, the set
$\sum_{[k]}A$ must also be an interval. Condition $(C)$ then follows since
adding back the sets $A_{k+1},\dots,A_{m}$ only translates the sets
$\sum_{T}A$. Finally, assume that each set $A_{1},\dots,A_{m}$ has
$\text{card}(A_{i})\geq 2$. Then each of the sets $\sum_{S}A$ contains two or
more points. By Lemma 4, each set $\sum_{S}A$ has positive measure. Condition
$(C)$ then follows from Theorem 3.

To prove the other direction, suppose that $(A)$ is true. Then equality is
immediately true with $0=0$. Next, suppose that condition $(B)$ is true. Then
the translated partition is the trivial partition which means that the
inequality is just $|\sum_{[k]}A|\geq|\sum_{[k]}A|$. It is clear that this
must be equality. Next assume that condition (C) is true. If all the sets
$\sum_{S}A$ have positive measure, then we use Theorem 3 to verify that
equality is true. Next, assume that at least one of the sets $\sum_{S}A$ has
zero measure. By assumption, those sets $\sum_{S}A$ with measure zero are
points. Then there exists some $k\in(1,m)$ such that the sets
$A_{1},\dots,A_{k}$ have $\text{card}(A_{i})\geq 2$ and the sets
$A_{k+1},\dots,A_{m}$ have $\text{card}(A_{i})=1$. Then the inequality can be
rewritten as

$\left|\sum_{[k]}A\right|\geq\sum_{T\in\mathcal{G}^{*}}\beta^{*}(T)\left|\sum_{T}A\right|.$

The sets $\sum_{T}A$ are all intervals of postive measure and the partition
$(\mathcal{G}^{*},\beta^{*})$ is non-trivial. Theorem 3 then verifies that
equality is true.

##  6 Appendix

###  6.1 The reduction to fractional partitions with rational weights

In proving the results of this work it has been assumed that the fractional
partitions have rational weights. In general this is not true. This appendix
will reiterate an argument given in [2] which justifies why this assumption is
sufficient.

The reduction to rational weights: Let $m\geq 3$ be an integer. The fractional
partitions of $[m]$ can be associated with sequences
$(\beta_{S})_{\varnothing\subsetneq S\subset[m]}$ such that for each
$i\in[m]$, $\sum_{S:i\in S}\beta_{S}=1$. We can ignore the term
$\beta_{\varnothing}$ since the empty set is irrelevant in the definition of a
fractional partition. The collection of all such sequences forms a convex,
compact polyhedral set $\mathcal{F}_{m}$:

$\mathcal{F}_{m}:=\left\\{(\beta_{S})_{\varnothing\subsetneq
S\subset[m]}:\forall S,0\leq\beta_{S}\leq 1\text{ and }\forall
i\in[m],\sum_{S:i\in S}\beta_{S}=1\right\\}.$

Let $\beta$ be an extreme point of $\mathcal{F}_{m}$. There exists a set
$K\subset 2^{[m]}$ such that $\beta_{S}>0$ for any $S\in K$ and $\beta_{S}=0$
for $S\notin K$. It will be shown that the extreme point $\beta$ is the unique
point in $\mathcal{F}_{m}$ such that $\beta_{S}>0$ for all $S\in K$ and for
all $i\in[m]$, $\sum_{S\in K:i\in S}\beta_{S}=1$. Suppose for contradiction
that there is another such $\beta^{\prime}\in\mathcal{F}_{m}$. Then
$\beta^{\prime}_{S}>0$ for $S\in K$ and $\beta^{\prime}_{S}=0$ for $S\notin
K$. Choose $\lambda>0$ so small that for each $S\in K$,
$\beta_{S}-\lambda\beta^{\prime}_{S}>0$. Now define a new set of weights by
$\beta^{\prime\prime}_{S}=(\beta_{S}-\lambda\beta^{\prime}_{S})/(1-\lambda)$
for $S\in K$ and $\beta^{\prime\prime}_{S}=0$ for $S\notin K$. It can be
verified that $\beta^{\prime\prime}\in\mathcal{F}_{m}$. Then for any $S\subset
2^{[m]}$,
$\beta_{S}=\lambda\beta^{\prime}_{S}+(1-\lambda)\beta^{\prime\prime}_{S}$.
Since $\beta$ is an extreme point of $\mathcal{F}_{m}$ and $\lambda\in(0,1)$
it must be that $\beta=\beta^{\prime}=\beta^{\prime\prime}$ which contradicts
the assumption that $\beta$ and $\beta^{\prime}$ are distinct. Therefore,
$\beta$ is the unique point of $\mathcal{F}_{m}$ which satisfies the system of
linear equations $\sum_{S\in K:i\in S}\beta_{S}=1$ for each $i\in[m]$. This
system has rational coefficients and by uniqueness is invertible. This implies
that the non-zero weights $\beta_{S}$ which give the solution must be
rational. Then the non-zero weights $\beta_{S}$ can be rewritten to have the
same denominator $q$. Accounting for repeats, this implies that a fractional
superadditive inequality for an extreme point $\beta$ can be written as
$q|A_{1}+\dots+A_{m}|\geq\sum_{j=1}^{s}|\sum_{S_{j}}A|$, where the $S_{j}$ are
an enumeration of the sets $S$ possibly with repetition and $q\geq 1$ is an
integer. Moreover, each $i\in[m]$ belongs to exactly $q$ of the sets $S_{j}$.
To see this, let $i\in[m]$. Then $i$ belongs to exactly $t$ sets in
$\mathcal{G}$ (not including repetition), denote them by
$S^{(1)},\dots,S^{(t)}$. Since the weights are rational with denominator $q$
we write

$\beta(S^{(j)})=\frac{n_{j}}{q},$

where the numbers $n_{j}$ are positive integers such that
$n_{1}+\dots+n_{t}=q$. If $i\in S^{(j)}$, the set $S^{(j)}$ is repeated
$n_{j}$ times, $j\in[t]$. Then $i$ belongs to at least $q$ sets (including
repetition), but cannot belong to any more. It follows that there are exactly
$q$ sets $S_{j}$ which contain $i$.

The equivalence of Theorem 4 for arbitrary fractional partitions: Let
$(\mathcal{G},\beta)$ represent some arbitrary fractional partiton of $[m]$.
It is a known fact that a bounded convex polytope is the convex hull of its
extreme points. Then there exist numbers $\alpha_{i}>0$, $i\in[N]$ with
$\sum_{i=1}^{N}\alpha_{i}=1$ and extreme points $\beta^{(i)}$ of
$\mathcal{F}_{m}$ such that
$\beta_{S}=\sum_{i=1}^{N}\alpha_{i}\beta^{(i)}_{S}$ for each
$S\in\mathcal{G}$. Suppose now that equality holds in the corresponding
fractional superadditive inequality. Then

$\begin{split}\left|\sum_{[m]}A\right|&=\sum_{S\in\mathcal{G}}\beta_{S}\left|\sum_{S}A\right|\\\
&=\sum_{i=1}^{N}\left(\alpha_{i}\sum_{S\in\mathcal{G}}\beta^{(i)}_{S}\left|\sum_{S}A\right|\right).\end{split}$

From this and the fact that
$|\sum_{[m]}A|=\sum_{i=1}^{N}\alpha_{i}|\sum_{[m]}A|$, it follows that
equality holds if and only if for each $i\in[k]$

$\left|\sum_{[m]}A\right|=\sum_{S\in\mathcal{G}}\beta^{(i)}_{S}\left|\sum_{S}A\right|.$

All that is left to show is that the same equality conditions of Theorem 4 are
also the same conditions for an arbitrary fractional partition. Using the
notation of Theorem 4, we only need to consider the case for when the
translated partition is a non-trivial partition of $[k]$. The translated
partition can be broken into a convex combination of rational partitions
(which also must be non-trivial partitions of $[k]$) as described above. If a
weight $\beta^{*}_{S}>0$, then one of the weights $\beta^{(i*)}_{S}$ for the
rational partitions is non-zero. By Theorem 4 for rational partitions, the
corresponding set $\sum_{S}A$ must be an interval or a point depending on its
volume. The reverse works the same: If the translated partition is non-
trivial, the sets $\sum_{S}A$ with positive measure are intervals, the sets
with zero measure are points, then break the partition into a convex
combination of rational partitions and apply Theorem 4.

###  6.2 Applying the Schneider non-convexity index to fractional partitions

As a technical detail in proving equality it was necessary to show (Lemma 2)
that for a fractional partition $(\mathcal{G},\beta)$ of $[m]$ and compact
sets $A_{1},\dots,A_{m}$ in $\mathbb{R}$ if for each $S\in\mathcal{G}$ the set
$\sum_{S}A$ is convex, then the set $\sum_{[m]}A$ is convex. This was shown by
using properties of compactness and the fractional superadditive inequalities
from Theorem 2. Due to the fact that in general dimension it is not true that
$|\text{conv}(A+B)|=|\text{conv}(A)|+|\text{conv}(B)|$ the technique used for
proving this fact in dimension $n=1$ does not work when $n>1$. The Schneider
non-convexity index [14] has a strong monotonicity property [8] which can be
used to prove this technical detail when the dimension is larger than 1. For a
compact set $A\subset\mathbb{R}^{n}$ the Schneider non-convexity index of $A$
is defined by

$c(A):=\inf\\{\lambda\geq 0:A+\lambda\text{conv}(A)\text{ is convex}\\},$

and has the property that if $A\subset\mathbb{R}^{n}$ is compact, $c(A)=0$ if
and only if $A$ is convex. In [8] it was shown that if $A_{1},\dots,A_{m}$ are
compact sets, and $S,T\subset[m]$ then the following inequality is true.

$c\left(\sum_{S\cup
T}A\right)\leq\max\left\\{c\left(\sum_{S}A\right),c\left(\sum_{T}A\right)\right\\}.$

If $(\mathcal{G},\beta)$ is a fractional partition of $[m]$, then there exist
sets $S^{(1)},\dots,S^{(t)}$ in $\mathcal{G}$ such that $S^{(1)}\cup\dots\cup
S^{(t)}=[m]$. This implies the following proposition.

Proposition 1. Let $m\geq 3$ be an integer and let $(\mathcal{G},\beta)$ be a
fractional partition of $[m]$. If $A_{1},\dots,A_{m}$ are compact sets in
$\mathbb{R}^{n}$ such that for any $S\in\mathcal{G}$ the set $\sum_{S}A$ is
convex, then the set $\sum_{[m]}A$ is convex.

Proof: By the above discussion, $c(\sum_{S^{(j)}}A)=0$ for each $j\in[t]$.
Using the above mentioned inequality it is true that for each $j<t$

$c\left(\sum_{(S^{(1)}\cup\dots S^{(j)})\cup
S^{(j+1)}}A\right)\leq\max\left\\{c\left(\sum_{S^{(1)}\cup\dots
S^{(j)}}A\right),c\left(\sum_{S^{(j+1)}}A\right)\right\\}.$

By induction the right hand side is equal to $0$ for each $j=1,\dots,t-1$.
Setting $j=t-1$ gives

$c\left(\sum_{[m]}A\right)=c\left(\sum_{S^{(1)}\cup\dots\cup
S^{(t)}}A\right)=0.$

This proves that the set $\sum_{[m]}A$ is convex.

###  6.3 An alternative approach to characterizing equality when the number of
sets is $m=3$

When characterizing equality for the case $m=3$ there is an alternate approach
which is worth mentioning. This approach is based on the observation that to
characterize equality for a fractional partition of $\\{1,2,3\\}$ we only need
to know the equality conditions for a few simple fractional partitions. To
simplify notation we will define the volumes $V_{i}$, $i\in[5]$ by

$\begin{split}V_{1}&:=\frac{1}{2}(|A_{12}|+|A_{13}|+|A_{23}|)\\\
V_{2}&:=|A_{12}|+|A_{3}|\\\ V_{3}&:=|A_{13}|+|A_{2}|\\\
V_{4}&:=|A_{23}|+|A_{1}|\\\ V_{5}&:=|A_{1}|+|A_{2}|+|A_{3}|.\end{split}$

In the next proposition, which is an explicit observation of what was proved
in Section 6.1 about extreme points, it will be shown that any fractional
superadditive inequality for $m=3$ depends on a convex combination of the
volumes $V_{i}$, $i\in[5]$.

Proposition 2. Let $A_{1},A_{2},A_{3}$ be non-empty compact subsets of
$\mathbb{R}^{n}$ and let $(\mathcal{G},\beta)$ be a fractional partition of
$\\{1,2,3\\}$. Then there exist non-negative real numbers $\alpha_{i}$, $1\leq
i\leq 5$ such that $\sum_{i=1}^{5}\alpha_{i}=1$ and

$\sum_{S\in\mathcal{G}}\beta(S)\left|\sum_{S}A\right|=\sum_{i=1}^{5}\alpha_{i}V_{i}.$

Proof: First observe that when $m=3$ we can write

$\sum_{S\in\mathcal{G}}\beta(S)\left|\sum_{S}A\right|=\beta_{12}|A_{12}|+\beta_{13}|A_{13}|+\beta_{23}|A_{23}|+\beta_{1}|A_{1}|+\beta_{2}|A_{2}|+\beta_{3}|A_{3}|.$

In this proof we drop the requirement made earlier that all the weights are
non-zero and allow the possiblility that some of the weights could equal zero.
By rearranging sets if needed we may assume that
$\beta_{3}=\min(\beta_{1},\beta_{2},\beta_{3})$. The choices of $\alpha_{i}$
are

$\begin{split}\alpha_{1}&=2\beta_{12}\\\ \alpha_{2}&=0\\\
\alpha_{3}&=\beta_{2}-\beta_{3}\\\ \alpha_{4}&=\beta_{1}-\beta_{3}\\\
\alpha_{5}&=\beta_{3}.\end{split}$

Then $\sum_{i=1}^{5}\alpha_{i}=1$, $\alpha_{i}\in[0,1]$ for each $i\in[5]$,
and $\sum_{S\in\mathcal{G}}|\sum_{S}A|=\sum_{i=1}^{5}\alpha_{i}V_{i}$.

Remark 3. Using the notation of Proposition 2 we first observe that for each
$i\in[5]$, $|A_{123}|\geq V_{i}$. Let $(\mathcal{G},\beta)$ be a fractional
partiton of $\\{1,2,3\\}$. Then by Proposition 2

$|A_{123}|=\sum_{S\in\mathcal{G}}\beta(S)\left|\sum_{S}A\right|$

if and only if $\sum_{i=1}^{5}\alpha_{i}(|A_{123}|-V_{i})=0$. This equality is
true if and only if $|A_{123}|=V_{i}$ for each $i\in[5]$ for which
$\alpha_{i}>0$. This shows that when $m=3$ we can find equality conditions for
any fractional partition as long as we know the equality conditions for the
equalities $|A_{123}|=V_{i}$.

## References

  * [1] Shiri Artstein, Keith Ball, Franck Barthe, and Assaf Naor.  Solution of shannon’s problem on the monotonicity of entropy.  Journal of the American Mathematical Society, 17(4):975–982, 2004\.
  * [2] Franck Barthe and Mokshay Madiman.  Volumes of subset Minkowski sums and le Lyusternik region.  arXiv e-prints, page arXiv:2112.06518, December 2021.
  * [3] SG Bobkov, Mokshay Madiman, and Liyao Wang.  Fractional generalizations of young and brunn-minkowski inequalities.  Concentration, functional inequalities and isoperimetry, 545:35–53, 2011.
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  * [5] W. R. Emerson and F. P. Greenleaf.  Asymptotic behavior of products $C^{p}=C+\cdots+C$ in locally compact abelian groups.  Trans. Amer. Math. Soc., 145:171–204, 1969.
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  * [9] Matthieu Fradelizi, Mokshay Madiman, and Artem Zvavitch.  Sumset estimates in convex geometry.  arXiv preprint arXiv:2206.01565, 2022.
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  * [11] LA Lusternik.  Die brunn-minkowskische ungleichung für beliebige messbare mengen.  CR Acad. Sci. URSS, 8:55–58, 1935.
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  * [17] Ross M Starr.  Quasi-equilibria in markets with non-convex preferences.  Econometrica: journal of the Econometric Society, pages 25–38, 1969\.


# Geant4Reweight: a framework for evaluating and propagating hadronic
interaction uncertainties in Geant4

J. Calcutt11footnotetext: Corresponding author. C. Thorpe  K. Mahn  L. Fields

###### Abstract

Geant4Reweight is an open-source C++ framework that allows users to weight
tracks produced by the Geant4 particle transport Monte Carlo simulation
according to hadron interaction cross section variations and estimate
uncertainties in Geant4 interaction models by comparing the simulation’s
hadron interaction cross section predictions to data. The ability to weight
hadron transport as simulated by Geant4 is crucial to the propagation of
systematic uncertainties related to secondary hadronic interactions in current
and upcoming neutrino oscillation experiments, including MicroBooNE, NOvA, and
DUNE, as well as hadron test beam experiments such as ProtoDUNE. We provide
motivation for weighting hadron tracks in Geant4 in the context of systematic
uncertainty propagation, a description of Geant4’s transport simulation
technique, and a description of our weighting technique and fitting framework
in the momentum range 0–10 GeV/c, which is typical for the hadrons produced by
neutrino interactions in these experiments.

##  1 Introduction

Measurements of neutrino oscillation parameters and neutrino-nucleus ($\nu$N)
cross sections performed by neutrino experiments, such as T2K [1], MicroBooNE
[2], NOvA [3], and the upcoming experiment DUNE [4], rely on determining the
energy, flavor, sign,222Whether it is a neutrino or antineutrino and
interaction topology of the detected neutrino event. However, these
experiments can only infer each of these through the identification of $\nu$N
interaction products. Issues in identifying these observables can arise in
various and subtle ways, many of which are associated with the re-interaction
in the detector of hadrons produced by the $\nu$N interaction — so called
“secondary interactions” (SI). A few examples are given below.

  1. 1.

Some experiments, such as NOvA and (in the future) DUNE, reconstruct the
neutrino energy via calorimetry of the neutrino interaction products. For
example, consider a $\nu_{\mu}$ Charged Current (CC) interaction of the form
$\nu_{\mu}+N\rightarrow\mu^{-}+p+\pi^{+}+N^{\prime}$. Here, the energy of the
hadronic system can be reconstructed using all non-muon energy depositions in
the detector and by identifying the $\pi^{+}$ track and including its rest
mass. The $\pi^{+}$ can undergo SI wherein it is absorbed by a nearby nucleus
in the detector i.e. $\pi^{+}+N\rightarrow N^{\prime}+2n$. In this case, a
majority of the identifiable kinetic energy of the $\pi^{+}$ as well as its
rest mass (139.57 MeV) would be lost.

  2. 2.

Experiments such as T2K use estimators for neutrino energy which rely only on
lepton kinematic information, and assume the event was a CCQE interaction
($\nu_{\mu}+p\rightarrow\mu^{-}+n)$ within a nucleus. For interactions wherein
a pion is produced, this energy estimate is biased. T2K identifies as CC0$\pi$
interactions events where there is no pion in the final state, and any number
of protons, to select a CCQE-enhanced event sample. Backgrounds within a
CC0$\pi$ selection include CC1$\pi$ events — which could arise from a resonant
interaction
($\nu_{\mu}+p\rightarrow\mu^{-}+\Delta^{++}\rightarrow\mu^{-}+p+\pi^{+}$) —
wherein the outgoing $\pi^{+}$ is absorbed nearby the neutrino interaction due
to SI.

  3. 3.

The signal for a $\nu_{e}$ CC event ($\nu_{e}+N\rightarrow
e^{-}+p+N^{\prime}$) includes an electromagnetic cascade produced by the final
state $e^{-}$ (also known as a “shower"). The electron in the $\nu_{e}$ CC
event is identified by the presence of one such shower. Backgrounds arise from
the presence of a $\pi^{0}$ in the final state, as this particle promptly
decays to photons that can produce similar showers. This $\pi^{0}$ could be
produced as a product of SI in the following interaction
$\nu_{\mu}+N\rightarrow\nu_{\mu}+p+\pi^{+}$ if the $\pi^{+}$ undergoes a
charge exchange interaction ($\pi^{+}+N\rightarrow\pi^{0}+N^{\prime}$) close
to the $\nu$N vertex.

  4. 4.

Consider a Neutral Current interaction of the form
$\bar{\nu}_{\mu}+N\rightarrow\bar{\nu}_{\mu}+p+\pi^{+}+N^{\prime}$. In the
case that the $\pi^{+}$ is particularly energetic and does not undergo SI, it
could mimic a $\mu$ as it comes to stop in the detector. The presence of the
apparent $\mu$ would potentially cause this event to be selected as a CC
event, meaning this event would contribute to a background.

Experiments typically correct for these effects in data using estimates from
Monte Carlo simulation. However, mismodeling of hadron SI rates will lead to
biases in this correction. An underprediction of hadron SI would underestimate
the neutrino energy (example 1) or overestimate the $\nu_{e}$-CC event rate
(example 3). An overestimation of $\pi^{+}$ SI would underestimate the rate of
backgrounds described in example 4. These examples highlight two needs from
experiments: 1) the ability to estimate uncertainties related to hadron SI and
2) the ability to propagate this uncertainty to signal and background event
rates.

Neutrino experiments typically rely on the Geant4 [5][6][7] toolkit to
simulate particles resulting from $\nu$N interactions as they pass through
detectors. This toolkit achieves this with the use of particle-nuclei cross
sections (to determine the rates of interactions occurring in the detector)
and physics models (to determine the specific dynamics of these interactions).
Prior to running, users choose which cross sections and models are used in the
simulation by selecting from standard or custom “physics lists”. Additionally,
certain model parameters can be varied before run-time. If users intend to
produce varied simulation results by choosing from a series of physics lists
or tweaking model parameters (i.e. for the purpose of uncertainty
propagation), they must rerun the full simulation and downstream processing
chain multiple times. This is computationally inefficient, especially when
considering any further simulation required for a neutrino experiment (i.e.
detector response and reconstruction). A common alternative to this is an
event-by-event weighting scheme which requires only one ensemble of Monte
Carlo events to be produced. This is commonly referred to as "reweighting."
When reweighting, the MC simulation is generated once under some nominal
model. The results of each simulated event are then considered under some
varied model, and weights are given to the events. When applied, the weights
will modify the population of events throughout regions of phase space
simulated using the varied model, thus approximating the results of rerunning
the simulation under the varied model. Independent implementations of hadronic
interaction reweighting have been used by recent neutrino experiments to
propagate hadron interaction uncertainties to their neutrino flux predictions
[8][9]. A similar weighting technique to that described in this paper has also
been used by T2K to propagate hadron SI model uncertainties [10]. In the
context of neutrino interactions, similar weighting schemes exist to propagate
systematic uncertainties on particle rescattering within the nuclear medium
(so-called Final State Interactions) [11][12].

The software package described here, Geant4Reweight [13], provides a framework
in which Geant4’s hadron interaction cross sections and models can be compared
to data in order to estimate uncertainties, which can then be applied to
analyses through reweighting. This framework is designed to be used in the
momentum range of 0–10 GeV/c which is typical of hadrons produced by neutrino
interactions in the aforementioned experiments. It is an open-source C++
framework that is intended to provide a unified approach across experiments to
estimating and propagating systematic uncertainties related to hadron SI. Many
internal methods rely on the ROOT [14] data analysis framework for storage or
fitting routines. The rest of this paper describes the following: section 2
focuses on the technique used by Geant4 to simulate particle transport through
material. This is described in order to inform the reweighting technique given
in section 3. Section 4 then provides examples of reweighting on simulated
$\pi^{+}$ tracks. Section 5 describes the fitting machinery used to estimate
uncertainties on Geant4 cross sections by comparing to external hadron
scattering data.

##  2 Geant4 simulation strategy

Before discussing the technical details of this reweighting framework, this
section provides a discussion of Geant4’s transport simulation. A detailed
description of Geant4’s simulation is available in Ref [15]. This section
provides a short description of the portion relevant to hadron SI reweighting.

Within Geant4’s simulation, a particle moves through the detector in an
iterative process whereby it takes a set of steps between points in space and
time. At each step, a set of processes (described further in the next
paragraph)333“Process” is the technical term for what is happening to the
particle during that step — i.e. it is simply moving between points or it is
undergoing an interaction. are considered and the simulation decides which
processes are invoked. The processes are categorized into three types (at-
rest, continuous, and discrete) that determine when in the step they occur,
and what criteria must be met before the process is invoked.444Some processes
are hybrid processes with combined behavior of multiple types of processes.
For example, decay processes are both discrete and at-rest. For our purposes,
the two behaviors can be treated separately If a particle stops, the at-rest
processes each propose an interaction time as an analog for an interaction
length, and the process that proposes the shortest time is invoked. When a
particle is not at rest, each continuous and discrete process proposes an
interaction length, and the shortest one is chosen as the step length.555If
the distance to the nearest volume boundary is shorter than the proposed
interaction length, it is chosen as the step size. Then, as the particle moves
between step end-points, each continuous process is invoked. Finally, at the
end of the step, if a discrete process proposed the shortest interaction
length, that process is invoked. For processes we are considering in the
context of the reweighting framework, the interaction lengths are provided by
cross section tables in the case of inelastic and elastic hadron scattering
and Coulomb scattering, and by lifetime tables in the case of pion decays. The
inelastic and elastic cross sections for $\pi^{+}$ interacting on argon are
shown in figure 1. These cross sections at this momentum range are taken from
calculations performed by Barashenkov [16] [17].

When a process is chosen to occur, Geant4 invokes an implementation of the
model corresponding to that process. The choice of model is customizable and
set based on the physics list chosen by the user. The results of the model are
a set of Geant4 tracks including the updated track (either with a change in
momentum direction and energy or it is removed from the stack of active
particles) after undergoing the process and any secondary tracks created by
the process. Geant4Reweight is intended to treat hadrons in the range of 0–10
GeV/c using physics lists commonly used by neutrino experiments. The physics
lists “G4HadronPhysicsQGSP_BERT” and “G4HadronElasticPhysics” are commonly
used by neutrino oscillation experiments at this momentum range for the
inelastic and elastic hadronic interactions, and are what we show in this
paper. For inelastic interactions, the Bertini Intranuclear Cascade model is
used for hadrons up to 10 GeV/c [16]. The differential elastic cross section
is calculated using the Glauber model [16][18] above 1 GeV/c and the Gheisha
model [19] below that.

Figure 1: Geant4 cross section predictions for $\pi^{+}$-Ar interactions.

##  3 Reweighting technique

The previous section provided a description of the Geant4 simulation strategy
and introduced relevant concepts such as processes, cross sections,
interaction lengths, and models in the parlance of Geant4’s simulation. This
section describes the reweighting technique used to create weights for
particle tracks under some given change to a process’ cross section. The
processes we are interested in reweighting (the hadronic elastic and inelastic
interactions) are discrete processes, so we will focus the remaining
discussion on this type of process. To inform the description of the
reweighting process, it is helpful to construct event probabilities that can
be varied according to the cross sections. The following discussion describes
this.

When the particle track is first created, each discrete process samples the
exponential distribution:

$p(N)=e^{-N}dN.$ (3.1)

This provides the process with a number of interaction lengths ($N$). The
interaction length proposed for a step is $N\lambda$, where $\lambda$ is the
mean free path of the process for that step and is inversely proportional to
the cross section for that process. At each step, if the process in question
did not occur, the process subtracts an amount equal to $n=\Delta L/\lambda$
from $N$ where $\Delta L$ is the step length taken and $n$ is the number of
interaction lengths taken by the step. Once this process is invoked in the
procedure detailed in section 2, $N$ is sampled again. Thus, for a given
process, the probability for that process to not occur after a series of steps
($P_{S}$ in short for $P_{Survive}$) is given by the probability to sample $N$
greater than the sum of all interaction lengths traveled by particle
($N_{T}$):

$\ \begin{split}P_{S}&=P(N>N_{T})\\\
&=\int_{N_{T}}^{\infty}e^{-N}dN=e^{-N_{T}}=e^{-\sum\limits_{i}^{f}n_{i}}\\\
&=e^{-\sum\limits_{i}^{f}\Delta
L_{i}/\lambda_{i}}=e^{-\sum\limits_{i}^{f}\Delta L_{i}\sigma_{i}}.\end{split}$
(3.2)

where $n_{i}=\Delta L_{i}/\lambda_{i}$ is the number of interaction lengths
traveled in step $i$. The last term uses the cross section
$\sigma_{i}=1/\lambda_{i}$ to simplify notation. Normally,
$1/\lambda=\sigma\rho N_{A}/M$, where $M$ and $\rho$ are the molar mass and
density of the surounding material, and $N_{A}$ is Avogadro’s number. The
extra factors have been absorbed into $\sigma$ for clarity. Note that the
cross section $\sigma_{i}$ is generally dependent on the particle’s momentum
and the surrounding material for step $i$. This probability can be rewritten
as the product of the probabilities for each step to occur without
interaction:

$P_{S}=\prod\limits_{i}^{f}P_{S,i}=\prod\limits_{i}^{f}e^{-\Delta
L_{i}\sigma_{i}}.$ (3.3)

If the process does occur, it only occurred in the last step, so the
probability ($P_{I}$ in short for $P_{Interact}$) is largely unchanged except
for the term contributing to the final step:

$\begin{split}P_{I}&=\bigg{(}\prod\limits_{i}^{f-1}P_{S,i}\bigg{)}\bigg{(}P_{I,f}\bigg{)}=\bigg{(}\prod\limits_{i}^{f-1}P_{S,i}\bigg{)}\bigg{(}1-P_{S,f}\bigg{)}\\\
&=\bigg{(}\prod\limits_{i}^{f-1}e^{-\Delta
L_{i}\sigma_{i}}\bigg{)}\bigg{(}1-e^{-\Delta
L_{f}\sigma_{f}}\bigg{)}.\end{split}$ (3.4)

These probabilities can be easily extended to consider multiple processes. In
equations 3.3 and 3.4, the cross sections will be extended to be sums of the
cross sections of all processes.

$\sigma=\sigma_{I}+\sigma_{E}+\sigma_{X}$ (3.5)

For our purposes, we are interested in reweighting the elastic and inelastic
hadronic interactions, so the cross sections for these processes are
explicitly stated in equation 3.5 as $\sigma_{I}$ and $\sigma_{E}$
respectively. All other processes are combined into a single effective cross
section $\sigma_{X}$. The processes included in $\sigma_{X}$ are Coulomb
scattering and decay (for the relevant particles) which is effectively treated
as an interaction with an interaction length given by $1/(\gamma\beta c\tau)$.
Thus, for a given process $p$, the probability it will occur is given by

$P_{I,p}=\bigg{(}\prod\limits_{i}^{N_{f}-1}e^{-\Delta
L_{i}\sigma_{i}}\bigg{)}\bigg{(}1-e^{-\Delta
L_{f}\sigma_{f}}\bigg{)}\bigg{(}\frac{\sigma_{p,f}}{\sigma_{f}}\bigg{)}.$
(3.6)

###  3.1 Weight calculations

The weights are calculated by considering a change to the underlying cross
sections and then determining whether the event was more or less probable
under that change. Geant4Reweight has the ability to reweight both the
inelastic and elastic hadronic cross sections ($\sigma_{I}$ and $\sigma_{E}$
respectively). We consider general variations to the inelastic and elastic
cross sections as such:

$\begin{split}\sigma_{I}&\rightarrow\sigma_{I}^{\prime}=f_{I}\sigma_{I}\\\
\sigma_{E}&\rightarrow\sigma_{E}^{\prime}=f_{E}\sigma_{E}.\end{split}$ (3.7)

A weight is given to an entire track by considering each step, creating a
weight for each step, and then multiplying these together:

$W_{Track}=\prod\limits_{i}^{f}W_{i}.$ (3.8)

For each step, one of four things can happen:

  1. 1.

The particle undergoes no interaction

  2. 2.

The particle undergoes an inelastic interaction

  3. 3.

The particle undergoes an elastic interaction

  4. 4.

The particle undergoes some other interaction

Each of these cases have different forms of weights. For the first, the weight
is given by the change in the survival probability for the step:

$\begin{split}W_{S,i}&=e^{-\Delta
L_{i}(\sigma_{I,i}^{\prime}+\sigma_{E,i}^{\prime}+\sigma_{X,i})}/e^{-\Delta
L_{i}(\sigma_{I,i}+\sigma_{E,i}+\sigma_{X,i})}\\\ &=e^{-\Delta
L_{i}(\sigma_{I,i}^{\prime}+\sigma_{E,i}^{\prime})}/e^{-\Delta
L_{i}(\sigma_{I,i}+\sigma_{E,i})}\end{split}.$ (3.9)

The other three weights are given by changes to interacting probabilities for
that step, as in the last two terms in equation 3.6.

$\begin{split}W_{I/E,i}&=\bigg{(}\frac{\sigma_{I/E,i}^{\prime}}{\sigma_{i}^{\prime}}\bigg{)}\bigg{(}\frac{\sigma_{i}}{\sigma_{I/E,i}}\bigg{)}\bigg{(}\frac{1-e^{-\Delta
L_{i}\sigma_{i}^{\prime}}}{1-e^{-\Delta L_{i}\sigma_{i}}}\bigg{)}\\\
&=f_{I/E}\bigg{(}\frac{\sigma_{i}}{\sigma_{i}^{\prime}}\bigg{)}\bigg{(}\frac{1-e^{-\Delta
L_{i}\sigma_{i}^{\prime}}}{1-e^{-\Delta L_{i}\sigma_{i}}}\bigg{)}\end{split}$
(3.10)
$\begin{split}W_{X,i}&=\bigg{(}\frac{\sigma_{X,i}}{\sigma_{i}^{\prime}}\bigg{)}\bigg{(}\frac{\sigma_{i}}{\sigma_{X,i}}\bigg{)}\bigg{(}\frac{1-e^{-\Delta
L_{i}\sigma_{i}^{\prime}}}{1-e^{-\Delta L_{i}\sigma_{i}}}\bigg{)}\\\
&=\bigg{(}\frac{\sigma_{i}}{\sigma_{i}^{\prime}}\bigg{)}\bigg{(}\frac{1-e^{-\Delta
L_{i}\sigma_{i}^{\prime}}}{1-e^{-\Delta L_{i}\sigma_{i}}}\bigg{)}\end{split}$
(3.11)

Here, $\sigma_{i}=\sigma_{I,i}+\sigma_{E,i}+\sigma_{X,i}$ and
$\sigma_{i}^{\prime}=\sigma_{I,i}^{\prime}+\sigma_{E,i}^{\prime}+\sigma_{X,i}$.

###  3.2 Exclusive channels

In neutrino experiments it is often important to consider not only the total
(also called inclusive) inelastic hadronic interactions but also exclusive
interactions. The examples listed in section 1 highlight this. For examples 1
and 2, the instigating interaction is $\pi^{+}$ absorption. In example 3, it
is $\pi^{+}$ charge exchange. However, in Geant4 the concept of exclusive
hadronic interactions is not explicitly defined, and the only accessible
information is the outgoing particles and their kinematics after a model is
invoked. In this way, the model can be thought of as a “black box” that
determines the results of the interaction (outgoing particles) which are then
added to the simulation stack and whose information users are able to access.
Therefore, Geant4Reweight defines exclusive interaction channels by the
outgoing particles following an interaction. For pions, the exclusive channel
names and their definitions are given in table 1.

Table 1:  Summary of exclusive channel definitions for charged pions. $X$ represents any number of nucleons in the final state. $n\pi$ represents more than one $\pi$ (any charge) in the final state. Channel Name | Definition
---|---
Quasielastic | $\pi^{\pm}+N\rightarrow\pi^{\pm}+N^{\prime}+X$
Absorption | $\pi^{\pm}+N\rightarrow N^{\prime}+X$
Charge Exchange | $\pi^{\pm}+N\rightarrow\pi^{0}+N^{\prime}+X$
Double Charge Exchange | $\pi^{\pm}+N\rightarrow\pi^{\mp}+N^{\prime}+X$
Production | $\pi^{\pm}+N\rightarrow n\pi+N^{\prime}+X$

For neutrino oscillation experiments such as those listed previously, the
model of choice is the Bertini Cascade model. In this model, the incident
hadron is injected into the nucleus and allowed to take a series of steps
within the nuclear medium. At each step, the particle has a chance to interact
with a particle within the nucleus and add it to the stack of active
particles. These then step through the nuclear matter, possibly interacting
along the way and activating more particles. This so-called cascade of
particles continues until all active particles stop within or exit the nucleus
and are added to the transport simulation outside of the nucleus. The latter
are the interaction results that users can access and which Geant4Reweight
uses to define the exculsive interaction channel. Geant4Reweight includes an
application which produces effective exclusive cross sections from the Bertini
Cascade that are used to calculate weights for exclusive interactions. It does
this by invoking the Bertini Cascade model many times over a given momentum
range to determine the fraction of times it ends in a specific final state.
This fraction is then multiplied by the inelastic cross section to create
exclusive cross sections. This is demonstrated in figure 2, where the
exclusive channel fractions are shown on the left and the exclusive cross
sections are shown on the right along with the total inelastic cross section.
These effective exclusive cross sections can be varied and reweighted in a
similar way to the full inelastic cross section.

(a) Exclusive final state fractions.

(b) Exclusive & Total Inelastic cross sections

Figure 2: Bertini Cascade model predictions and Geant4 cross section
predictions for inelastic $\pi^{+}$-Ar interactions. The exclusive cross
sections on the right are produced by multiplying the fractions on the left by
the total inelastic cross section on the right.

The reweighting scheme is easily extended to include the exclusive cross
sections. We treat the inelastic cross section as the sum over multiple
exclusive channels:

$\sigma_{I}=\sum\limits_{c}^{N_{c}}\sigma_{c}.$ (3.12)

We allow users to specify a variation factor to each of the exclusive channels
as well as to the total inelastic cross section similar to before:

$\sigma_{I}\rightarrow\sigma_{I}^{\prime}=f_{I}\bigg{(}\sum\limits_{c}^{N_{c}}f_{c}\sigma_{c}\bigg{)}.$
(3.13)

$W_{S}$, $W_{E}$, and $W_{X}$ remain largely the same with $\sigma^{\prime}$
and $\sigma_{I}^{\prime}$ changed accordingly. $W_{I}$, however, is modified
to account for the separate variations to the exclusive channels. Here, we
also have to account for the exclusive channel ($j$) of the interaction. For
clarity, the subscript $i$ denoting that this occurs for a given step has been
omitted.

$\begin{split}W_{I,j}&=\bigg{(}\frac{f_{j}\sigma_{j}}{\sum\limits_{c}^{N_{c}}f_{c}\sigma_{c}}\bigg{)}\bigg{(}\frac{\sigma_{I}}{\sigma_{j}}\bigg{)}\bigg{(}\frac{f_{I}\sum\limits_{c}^{N_{c}}f_{c}\sigma_{c}}{\sigma^{\prime}}\bigg{)}\bigg{(}\frac{\sigma}{\sigma_{I}}\bigg{)}\bigg{(}\frac{1-e^{-\Delta
L\sigma^{\prime}}}{1-e^{-\Delta L\sigma}}\bigg{)}\\\
&=f_{j}f_{I}\bigg{(}\frac{\sigma}{\sigma^{\prime}}\bigg{)}\bigg{(}\frac{1-e^{-\Delta
L\sigma^{\prime}}}{1-e^{-\Delta L\sigma}}\bigg{)}\end{split}$ (3.14)

##  4 Reweighting examples

In this section, we present examples of validation of Geant4Reweight’s
weighting scheme applied to simulated particle tracks. Samples of $\pi^{+}$
were generated using a particle gun in the liquid argon of the ProtoDUNE-SP
detector [20][21] using the LArSoft toolkit [22]. The samples had a Gaussian
momentum profile centered at $1~{}\textrm{GeV/c}$ with a width of
$100~{}\textrm{MeV/c}$. Within LArSoft’s interface to Geant4, custom Geant4
physics lists were written to allow us to separately vary the inelastic and
elastic $\pi^{+}$ cross sections. The generated hadrons were then propagated
through liquid argon using the custom physics lists under three
configurations:

  1. 1.

Nominal inelastic and elastic cross sections

  2. 2.

Inelastic cross section multiplied by 1.5. Nominal elastic cross section.

  3. 3.

Elastic cross section multiplied by 1.5. Nominal inelastic cross section.

The results of this simulation were then passed to a LArSoft module that
translated the LArSoft-style data products into Geant4Reweight objects. The
nominal simulation results were then reweighted according to the varied
configurations above.

###  4.1 Changing the $\pi^{+}$ inelastic cross section

Figure 3 shows comparison plots for the following samples: 1) running Geant4
with both of the nominal cross sections (“Nominal” in the plots), 2) running
Geant4 with the inelastic cross section multiplied by 1.5 (“Varied”), and 3)
applying Geant4Reweight to the nominal cross section sample according to the
inelastic cross section change (“Weighted”). In figure 3(a), the track length
of the particle is shown. This is the integrated path length traveled by the
$\pi^{+}$ from its creation to the point at which it stops, decays in flight,
or undergoes an inelastic interaction. Increasing the inelastic cross section
increases the chance that it will interact earlier, before stopping or
decaying, thus shifting the Weighted and Varied distributions lower, as
expected. Figure 3(b) shows the number of elastic scatters performed by the
$\pi^{+}$ during its travel. Similarly, increasing the inelastic cross
sections means the $\pi^{+}$ will be less likely to travel as far as in the
nominal simulation, suppressing the chance for elastic scatters to occur. The
figures show good agreement between the varied and weighted samples,
supporting the validity of our reweighting scheme.

(a)

(b)

Figure 3: Comparison of pion track lengths (left) and number of elastic
scatters (right) using Geant4 with nominal cross sections (Nominal) and with
inelastic cross sections varied directly in Geant4 (Varied) and using
Geant4Reweight (Weighted).

###  4.2 Changing the $\pi^{+}$ elastic cross section

Figure 4 shows similar comparison plots as the previous section, but for the
sample with the elastic cross section multiplied by 1.5 and the inelastic
cross section kept the same. In figure 4(a), the track length distribution
shows no significant change. This is due to the track-length definition above,
as well as the fact that the elastic interaction in Geant4 removes minimal
energy from the $\pi^{+}$. As such, more elastic scatters do not cause the
$\pi^{+}$ to stop earlier. Figure 4(b) shows that the number of elastic
scatters performed by the $\pi^{+}$ increases as expected.

(a)

(b)

Figure 4: Comparison of pion track lengths (left) and number of elastic
scatters (right) using Geant4 with nominal cross sections (Nominal) and with
elastic cross sections varied directly in Geant4 (Varied) and using
Geant4Reweight (Weighted).

###  4.3 Recovering the nominal cross section

As a final demonstration, we show the ability for Geant4Reweight to recover
the nominal cross sections in Geant4 by reweighting the samples generated with
varied cross sections. This is shown first in figure 5. Here, “Nominal” and
“Varied” are samples generated with nominal cross sections and with the
inelastic cross section scaled by 1.5. “Weighted*” shows a reweighting applied
to the Varied sample, where a scaling factor of 2/3 is applied to the varied
cross section to return to the nominal. The figure shows the ability to
reweight from a varied cross section back to nominal.

Figure 6 shows a similar treatment, but with the elastic cross section
increased by 50% and then weighted to recover the nominal model. The nominal
elastic cross section can be recovered via reweighting as shown.

(a)

(b)

Figure 5: Comparison of pion track lengths (left) and number of elastic
scatters (right) using Geant4 with nominal cross sections (Nominal) and with
inelastic cross sections varied directly in Geant4 (Varied) and using
Geant4Reweight applied to the Varied sample (Weighted*).

(a)

(b)

Figure 6: Comparison of pion track lengths (left) and number of elastic
scatters (right) using Geant4 with nominal cross sections (Nominal) and with
elastic cross sections varied directly in Geant4 (Varied) and using
Geant4Reweight applied to the Varied sample (Weighted*).

##  5 Fitting framework

Geant4Reweight includes a fitting framework to estimate uncertainties on the
Geant4 cross sections. These uncertainties can then be propagated through
physics analysis via reweighting. Sets of data are provided by Geant4Reweight,
and users are able to include additional data as well. This framework uses the
MIGRAD algorithm from the MINUIT package implemented within ROOT. Users
provide a configuration file defining the following:

  1. 1.

The input final state fraction files. These are used as in section 3.2 to
define exclusive cross sections.

  2. 2.

The data sets against which the Geant4 predictions are fit. These include the
following:

    1. (a)

The path to the ROOT file containing the data points

    2. (b)

The incident particle type and target material.

    3. (c)

The category of cross sections within the data file (i.e. Total Inelastic,
Absorption, etc.).

    4. (d)

The name of the ROOT objects representing the cross section data points within
the data file.

  3. 3.

A set of parameters which represent variations to the cross sections across
defined regions of momentum.

  4. 4.

Configurations for the fit minimizer.

###  5.1 Parameter definitions

The parameters used within the fit are user-defined variations across regions
of incident hadron momentum, for the various cross sections. For charged
pions, this includes the exclusive cross sections defined in table 1, as well
as the total inelastic (the sum of the exclusive channels — also referred to
as reaction) and elastic cross sections. The reaction cross section parameters
work simultaneously with the exclusive parameters, similar to the weighting
scheme in section 3. Flat variations are applied across the momentum range
chosen by users.

During each iteration of the fit, the values of the user-defined parameters
are changed. The cross sections are scaled according to the variations defined
by the new parameter values, and fit to the provided data. If the fit is
successful, the best-fit values of the parameters and their covariances are
saved in the output ROOT file.

###  5.2 $\chi^{2}$ definitions

The data in the fit is categorized by the interaction channel of each
measurement. The relevant channels are the exclusive channels (such as those
listed in table 1), combinations of these (such as pion absorption + charge
exchange), the elastic interaction, the reaction (total inelastic)
interaction, and the inclusive interaction (reaction + elastic). The quantity
of measurements for each exclusive channel varies widely, and a simple
$\chi^{2}$ calculation will likely lead to channels with more measurements
dominating the fit. Instead, a reduced $\chi^{2}$ statistic is calculated for
each channel, as shown in equation 5.1. Here, $k$ denotes the interaction
channel, $N_{k}$ is the number of cross section measurements for channel $k$,
and $p_{k}$ is the number of user-defined parameters for that channel.
$\sigma^{\textrm{MC}}$ denotes the Geant4 cross section that has been varied
by the fit parameters as described above.

$\displaystyle\chi^{2}_{k}$
$\displaystyle=\frac{1}{N_{k}-p_{k}}\sum_{i}^{N_{k}}\left(\frac{\sigma^{\textrm{
Data}}_{i}-\sigma^{\textrm{MC}}_{i}}{\Delta\sigma^{\textrm{Data}}_{i}}\right)^{2}$
(5.1) $\displaystyle\chi^{2}$
$\displaystyle=\frac{1}{N_{\textrm{C}}+f-1}\bigg{(}\sum\limits_{k}\chi^{2}_{k}+f\chi^{2}_{\textrm{Total}}\bigg{)}$
(5.2)

The full $\chi^{2}$ is shown in equation 5.2. Here, the $\chi^{2}$
contributions from all channels other than the inclusive interaction are
simply summed. The inclusive interaction is treated slightly differently
because variations to the total cross section can sometimes dominate the fit.
As such, a scaling parameter $f$ is introduced, allowing the user to control
the contribution of the inclusive channel to the full $\chi^{2}$. The leading
factor is the effective number of degrees of freedom, where $N_{C}$ is the
number of channels included in the sum. This term ensures $\chi^{2}\sim 1$ as
$\big{(}\sigma^{\textrm{ Data}}_{i}-\sigma^{\textrm{MC}}_{i}\big{)}$
approaches $\Delta\sigma^{\textrm{Data}}$.

####  5.2.1 Fit with correlated data

The reduced $\chi^{2}$ statistic defined in equation 5.1 is valid for
uncorrelated data. However, a recent measurement by the DUET collaboration
includes correlations in their published results for $\pi^{+}$ absorption and
charge exchange on carbon [23]. We have included this data in the fit in a
specific way to account for these correlations. A standard $\chi^{2}$
definition is shown in equation 5.3. For this set of data, the absorption and
charge exchange cross sections were each measured at five momenta. As such,
the sum in this equation runs over all ten data points. In order to treat the
measurements of these two channels in a similar manner as above, we have
chosen to split this value equally between the modified reduced $\chi^{2}$
statistics for the absorption and charge exchange channels, as shown in
equations 5.4 and 5.5. The denominator of the first term in each of these
equations includes the five data points for each channel.

$\chi^{2}_{\textrm{DUET}}=\sum\limits_{i,j}^{10}\big{(}\sigma^{\textrm{DUET}}_{i}-\sigma^{\textrm{MC}}_{i}\big{)}V_{ij}^{-1}\big{(}\sigma^{\textrm{DUET}}_{j}-\sigma^{\textrm{MC}}_{j}\big{)}$
(5.3)

$\chi^{2}_{\textrm{Abs}}=\frac{1}{N_{\textrm{Abs}}+5-p_{\textrm{Abs}}}\Bigg{(}\sum\limits^{N_{\textrm{Abs}}}_{i}\bigg{(}\frac{\sigma_{i}^{\textrm{Data}}-\sigma_{i}^{\textrm{MC}}}{\Delta\sigma^{\textrm{Data}}_{i}}\bigg{)}^{2}+\frac{\chi^{2}_{\textrm{DUET}}}{2}\Bigg{)}$
(5.4)

$\chi^{2}_{\textrm{Cex}}=\frac{1}{N_{\textrm{Cex}}+5-p_{\textrm{Cex}}}\Bigg{(}\sum\limits^{N_{\textrm{Cex}}}_{i}\bigg{(}\frac{\sigma_{i}^{\textrm{Data}}-\sigma_{i}^{\textrm{MC}}}{\Delta\sigma^{\textrm{Data}}_{i}}\bigg{)}^{2}+\frac{\chi^{2}_{\textrm{DUET}}}{2}\Bigg{)}$
(5.5)

In general, other correlated data can be included in the fit in a similar
manner. If such a set of data exists, a user can implement their own
calculation of its contribution to the specific $\chi^{2}_{k}$.

###  5.3 Example fit

For demonstration, we have fit Geant4 to a collection of $\pi^{+}$-C data. The
parameters we have chosen to use are shown in table 2. The data included in
the fit is shown in table 3. Throughout this section, the “reaction” channel
is used as another term for any inelastic pion-nucleus interaction. Its cross
section is the combination of all exclusive channels (i.e. it is the total
inelastic cross section shown in figure 1).

Table 2:  The parameters used within the demonstration fit. Parameter Name | Channel | Momentum Range [MeV$/c$]
---|---|---
fReacLow | Reaction | 10–200
fReacHigh | Reaction | 700–2005
fAbs | Absorption | 200–700
fCex | Charge Exchange | 200–700
fInel | Quasielastic | 200–700
Table 3:  The data sets used within the demonstration fit. Reference | Channel(s) | Momentum Range [MeV/$c$]
---|---|---
B. W. Allardyce et al. [24] | Reaction | 710–2000
D. Ashery et al. [25] | Quasielastic, Abs. + Ch. Ex. | 175–433
D. Ashery et al. [26] | Charge Exchange | 265
E. Bellotti et al. [27] | Absorption, Charge Exchange | 230
S. M. Levenson et al. [28] | Quasielastic | 194–417
O. Meirav et al. [29] | Reaction | 128–170
I. Navon et al. [30] | Abs. + Ch. Ex. | 128
E. S. Pinzon-Guerra et al. [23] | Absorption, Charge Exchange | 206–295
A. Saunders et al. [31] | Reaction | 116–150

The results are shown in figures 7–9. Figure 7 shows the best fit values for
the parameters and their covariance. Of note are the slightly larger
covariances between the absorption and charge exchange parameters which are
introduced due to including the combined absorption and charge exchange cross
section data. In figure 8, cross section data used in the fit is compared to
the nominal Geant4 prediction (blue line) and the best fit (black line) & $\pm
1\sigma$ error bands (red area). These figures highlight a current limitation
in the framework as seen in the discontinuities in the best fit values of the
cross sections and the error band sizes at the parameter transitions. Highly
precise reaction data is included for momentum above 700 MeV/$c$. As such,
this reduces the error on the high-momentum parameter as compared to the
lower-momentum regions on the parameter affecting that region when compared to
the middle region, which is fit to less accurate data. Finally, in figure 9
are 1-D parameter scans of the fit $\chi^{2}$ around the best-fit point. All
plots are generated by the fitting application and included in the output.

(a) Best-fit parameter values and 1-D uncertainty.

(b) Output covariance matrix.

Figure 7:

(a)

(b)

(c)

(d)

(e)

Figure 8: Fit results of $\pi^{+}$-C cross sections extracted from Geant4
(with the Bertini Cascade model) fit to data (listed in table 3). The
parameters used in the fit are defined in table 2. The blue dashed line is the
nominal cross section when extracted from Geant4, the black dashed line is the
best fit value, and the red bands are calculated by setting the parameters to
their $\pm 1\sigma$ values.

(a) fReacLow

(b) fReacHigh

(c) fAbs

(d) fCex

(e) fInel

Figure 9: 1-D parameter scans of the fit $\chi^{2}$ near the best-fit point.

##  6 Summary

Geant4Reweight will serve the neutrino physics community by providing a
framework that allows users to estimate and propagate systematic uncertainties
related to hadron–nucleus interactions in Geant4. This paper describes the
framework and its validation. Multiple experiments — including MicroBooNE,
NOvA, and ProtoDUNE-SP — have begun to use this framework for uncertainty
propagation in their analyses.

## Acknowledgments

This material is based upon work supported by the U.S. Department of Energy
(DOE), Office of Science, Office of Workforce Development for Teachers and
Scientists, Office of Science Graduate Student Research (SCGSR) program. The
SCGSR program is administered by the Oak Ridge Institute for Science and
Education for the DOE under contract number DE-SC0014664. Authors are also
supported by DOE Award Number DE-SC0015903. We acknowledge the hard work of
the Geant4 collaboration in the creation, documentation, and maintenance of
the Geant4 software package, without which this work would not be possible. We
also gratefully acknowledge a careful reading of this manuscript and useful
comments by Wayne Repko.

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