1.1 An Informal Introduction

A voting method or social choice function with m candidates and n voters is a function

$$ \begin{align*}f\colon\{1,\ldots,m\}^{n}\to\{1,\ldots,m\}.\end{align*} $$

From the social choice theory perspective, the input of the function f is a list of votes of n people who are choosing between m candidates. Each of the m candidates is labelled by the integers $1,\ldots ,m$. If the votes are $x\in \{1,\ldots ,m\}^{n}$, then $x_{i}$ denotes the vote of person $i\in \{1,\ldots ,n\}$ for candidate $x_{i}\in \{1,\ldots ,m\}$. Given the votes $x\in \{1,\ldots ,m\}^{n}$, $f(x)$ is interpreted as the winner of the election.

It is both natural and desirable to find a voting method whose output is most likely to be unchanged after votes are randomly altered. One could imagine that malicious third parties or miscounting of votes might cause random vote changes, so we desire a voting method f whose output is stable to such changes. In addition to voting motivations, finding a voting method that is stable to noise has applications to the unique games conjecture [Reference Khot, Kindler, Mossel and O’DonnellKKMO07, Reference Mossel, O’Donnell and OleszkiewiczMOO10, Reference Khot and MoshkovitzKM16], to semidefinite programming algorithms such as MAX-CUT [Reference Khot, Kindler, Mossel and O’DonnellKKMO07, Reference Isaksson and MosselIM12], to learning theory [Reference Feldman, Guruswami, Raghavendra and WuFGRW12], etc. For some surveys on this and related topics, see [Reference O’DonnellO’D, Reference KhotKho, Reference HeilmanHei20].

The output of a constant function f is never altered by changes to the votes. Also, if the function f only depends on one of its n inputs, then the output of f is rarely changed by independent random changes to each of the votes. In these cases, the function f is rather ‘undemocratic’ from the perspective of social choice theory. In the case of a constant function, the outcome of the election does not depend at all on the votes. In the case of a function that only depends on one of its inputs, the outcome of the election only depends on one voter (so f is called a dictatorship function).

Among ‘democratic’ voting methods, it was conjectured in [Reference Khot, Kindler, Mossel and O’DonnellKKMO07] and proven in [Reference Mossel, O’Donnell and OleszkiewiczMOO10] that the majority voting method is the voting method that best preserves the outcome of the election. The following is an informal statement of the main result of [Reference Mossel, O’Donnell and OleszkiewiczMOO10].

We say a vote $x_{i}\in \{1,2\}$ is corrupted with probability $0<\delta <1$ when, with probability $\delta $, the vote $x_{i}$ is changed to a uniformly random element of $\{1,2\}$, and with probability $1-\delta $, the vote $x_{i}$ is unchanged.

For a formal statement of Theorem 1.1, see Theorem 1.8 below.

The primary interest of the authors of [Reference Khot, Kindler, Mossel and O’DonnellKKMO07] in Theorem 1.1 was proving optimal hardness of approximation for the MAX-CUT problem. In the MAX-CUT problem, we are given a finite undirected graph on n vertices, and the objective of the problem is to find a partition of the vertices of the graph into two sets that maximises the number of edges going between the two sets. The MAX-CUT problem is MAX-SNP hard, i.e. if $P\neq NP$, there is no polynomial time (in n) approximation scheme for this problem. Nevertheless, there is a randomised polynomial time algorithm [Reference Goemans and WilliamsonGW95] that achieves, in expectation, at least $.87856\ldots $ times the maximum value of the MAX-CUT problem. This algorithm uses semidefinite programming. Also, the exact expression for the $.87856\ldots $ constant is

$$ \begin{align*} \min_{-1\leq\rho\leq 1}\frac{2}{\pi}\frac{\arccos(\rho)}{1-\rho}=.87856\ldots \end{align*} $$

The authors of [Reference Khot, Kindler, Mossel and O’DonnellKKMO07] showed that, if the Unique Games Conjecture is true, then Theorem 1.1 implies that the Goemans-Williamson algorithm’s .87856$\ldots $ constant of approximation cannot be increased. Assuming the validity of the Unique Games Conjecture is a fairly standard in complexity theory, though the conjecture remains open. See [Reference O’DonnellO’D, Reference KhotKho] and the references therein for more discussion on this conjecture, and see [Reference Khot, Minzer and SafraKMS18] for some recent significant progress.

Theorem 1.1 (i.e. Theorem 1.8) gives a rather definitive statement on the two candidate voting method that is most stable to corruption of votes. Moreover, the applcation of Theorem 1.1 gives a complete understanding of the optimal algorithm for solving MAX-CUT, assuming the Unique Games Conjecture. Unfortunately, the proof of Theorem 1.1 says nothing about elections with $m>2$ candidates. Moreover, Theorem 1.1 fails to prove optimality of the Frieze-Jerrum [Reference Frieze and JerrumFJ95] semidefinite programming algorithm for the MAX-m-CUT problem. In the MAX-m-CUT problem, we are given a finite undirected graph on n vertices, and the objective of the problem is to find a partition of the vertices of the graph into m sets that maximises the number of edges going between the two sets. So, MAX-CUT is the same as MAX-2-CUT.

In order to prove the optimality of the Frieze-Jerrum [Reference Frieze and JerrumFJ95] semidefinite programming algorithm for the MAX-m-CUT problem, one would need an analogue of Theorem 1.1 for $m>2$ voters, where the plurality function replaces the majority function. For this reason, it was conjectured [Reference Khot, Kindler, Mossel and O’DonnellKKMO07, Reference Isaksson and MosselIM12] that the plurality function is the voting method that is most stable to independent, random vote corruption.

We say that a vote $x_{i}\in \{1,\ldots ,m\}$ is corrupted with probability $0<\delta <1$ when, with probability $\delta $, the vote $x_{i}$ is changed to a uniformly random element of $\{1,\ldots ,m\}$, and with probability $1-\delta $, the vote $x_{i}$ is unchanged.

In the case that the probability of vote corruption goes to zero, the first author proved the first known case of Conjecture 1.2 in [Reference HeilmanHei19], culminating in a series of previous works [Reference Colding and MinicozziCM12, Reference McGonagle and RossMR15, Reference Barchiesi, Brancolini and JulinBBJ17, Reference HeilmanHei17, Reference Milman and NeemanMN18a, Reference Milman and NeemanMN18b, Reference HeilmanHei18]. Conjecture 1.2 for all fixed parameters $0<\rho <1$ was entirely open until now. Unlike the case of the majority is stablest (Theorem 1.8), Conjecture 1.2 cannot hold when the candidates have unequal chances of winning the election [Reference Heilman, Mossel and NeemanHMN16]. This realization is an obstruction to proving Conjecture 1.2. It suggested that existing proof methods for Theorem 1.8 cannot apply to Conjecture 1.2.

Nevertheless, we are able to overcome this obstruction in the present work.

Theorem 1.3 is the first proven case of the plurality is stablest conjecture (Conjecture 1.2).

1.2 More Formal Introduction

Using a generalization of the central limit theorem known as the invariance principle [Reference Mossel, O’Donnell and OleszkiewiczMOO10, Reference Isaksson and MosselIM12], there is an equivalence between the discrete problem of Conjecture 1.2 and a continuous problem known as the standard simplex conjecture [Reference Isaksson and MosselIM12]. For more details on this equivalence, see Section 7 of [Reference Isaksson and MosselIM12]. We begin by providing some background for the latter conjecture, stated in Conjecture 1.6.

For any $k\geq 1$, we define the Gaussian density as

(1)$$ \begin{align} \begin{aligned} \gamma_{k}(x)&\colon = (2\pi)^{-k/2}e^{-\left\|x\right\|^{2}/2},\qquad \langle x,y\rangle\colon=\sum_{i=1}^{n+1}x_{i}y_{i},\qquad \left\|x\right\|^{2}\colon=\langle x,x\rangle,\\ &\qquad\forall\,x=(x_{1},\ldots,x_{n+1}),y=(y_{1},\ldots,y_{n+1})\in\mathbb{R}^{n+1}. \end{aligned} \end{align} $$

Let $z_{1},\ldots ,z_{m}\in \mathbb {R}^{n+1}$ be the vertices of a regular simplex in $\mathbb {R}^{n+1}$ centred at the origin. For any $1\leq i\leq m$, define

(2)$$ \begin{align} \Omega_{i}\colon=\{x\in\mathbb{R}^{n+1}\colon\langle x,z_{i}\rangle=\max_{1\leq j\leq m}\langle x,z_{j}\rangle\}. \end{align} $$

We refer to any sets satisfying (2) as cones over a regular simplex.

Let $f\colon \mathbb {R}^{n+1}\to [0,1]$ be measurable and let $\rho \in (-1,1)$. Define the Ornstein–Uhlenbeck operator with correlation $\rho $ applied to f by

(3)$$ \begin{align} \begin{aligned} T_{\rho}f(x) \colon =\,&\int_{\mathbb{R}^{n+1}}f(x\rho+y\sqrt{1-\rho^{2}})\gamma_{n+1}(y)\,\mathrm{d} y\\ =\,&(1-\rho^{2})^{-(n+1)/2}(2\pi)^{-(n+1)/2}\int_{\mathbb{R}^{n+1}}f(y)e^{-\frac{\left\|y-\rho x\right\|^{2}}{2(1-\rho^{2})}}\,\mathrm{d} y, \qquad\forall x\in\mathbb{R}^{n+1}. \end{aligned} \end{align} $$

$T_{\rho }$ is a parametrization of the Ornstein–Uhlenbeck operator, which gives a fundamental solution of the (Gaussian) heat equation

(4)$$ \begin{align} \frac{d}{d\rho}T_{\rho}f(x)=\frac{1}{\rho}\Big(-\overline{\Delta} T_{\rho}f(x)+\langle x,\overline{\nabla}T_{\rho}f(x)\rangle\Big),\qquad\forall\,x\in\mathbb{R}^{n+1}. \end{align} $$

Here $\overline {\Delta }\colon =\sum _{i=1}^{n+1}\partial ^{2}/\partial x_{i}^{2}$ and $\overline {\nabla }$ is the usual gradient on $\mathbb {R}^{n+1}$. $T_{\rho }$ is not a semigroup, but it satisfies $T_{\rho _{1}}T_{\rho _{2}}=T_{\rho _{1}\rho _{2}}$ for all $\rho _{1},\rho _{2}\in (0,1)$. We have chosen this definition because the usual Ornstein–Uhlenbeck operator is only defined for $\rho \in [0,1]$.

Definition 1.4 Noise Stability

Let $\Omega \subseteq \mathbb {R}^{n+1}$ be measurable. Let $\rho \in (-1,1)$. We define the noise stability of the set $\Omega $ with correlation $\rho $ to be

$$ \begin{align*} \int_{\mathbb{R}^{n+1}}1_{\Omega}(x)T_{\rho}1_{\Omega}(x)\gamma_{n+1}(x)\,\mathrm{d} x \stackrel{(3)}{=}(2\pi)^{-(n+1)}(1-\rho^{2})^{-(n+1)/2}\int_{\Omega}\int_{\Omega}e^{\frac{-\|x\|^{2}-\|y\|^{2}+2\rho\langle x,y\rangle}{2(1-\rho^{2})}}\,\mathrm{d} x\mathrm{d} y. \end{align*} $$

Equivalently, if $X=(X_{1},\ldots ,X_{n+1}),Y=(Y_{1},\ldots ,Y_{n+1})\in \mathbb {R}^{n+1}$ are $(n+1)$-dimensional jointly Gaussian distributed random vectors with $\mathbb {E} X_{i}Y_{j}=\rho \cdot 1_{(i=j)}$ for all $i,j\in \{1,\ldots ,n+1\}$, then

$$ \begin{align*} \int_{\mathbb{R}^{n+1}}1_{\Omega}(x)T_{\rho}1_{\Omega}(x)\gamma_{n+1}(x)\,\mathrm{d} x=\mathbb{P}((X,Y)\in \Omega\times \Omega). \end{align*} $$

Maximising the noise stability of a Euclidean partition is the continuous analogue of finding a voting method that is most stable to random corruption of votes among voting methods where each voter has a small influence on the election’s outcome.

Problem 1.5 Standard Simplex Problem [Reference Isaksson and MosselIM12]

Let $m\geq 3$. Fix $a_{1},\ldots ,a_{m}>0$ such that $\sum _{i=1}^{m}a_{i}=1$. Fix $\rho \in (0,1)$. Find measurable sets $\Omega _{1},\ldots \Omega _{m}\subseteq \mathbb {R}^{n+1}$ with $\cup _{i=1}^{m}\Omega _{i}=\mathbb {R}^{n+1}$ and $\gamma _{n+1}(\Omega _{i})=a_{i}$ for all $1\leq i\leq m$ that maximise

$$ \begin{align*} \sum_{i=1}^{m}\int_{\mathbb{R}^{n+1}}1_{\Omega_{i}}(x)T_{\rho}1_{\Omega_{i}}(x)\gamma_{n+1}(x)\,\mathrm{d} x, \end{align*} $$

subject to the above constraints. (Here $\gamma _{n+1}(\Omega _{i})\colon =\int _{\Omega _{i}}\gamma _{n+1}(x)\,\mathrm {d} x \ \forall \ 1\leq i\leq m$.)

We can now state the continuous version of Conjecture 1.2.

Conjecture 1.6 Standard Simplex Conjecture [Reference Isaksson and MosselIM12]

Let $\Omega _{1},\ldots \Omega _{m}\subseteq \mathbb {R}^{n+1}$ maximise Problem 1.5. Assume that $m-1\leq n+1$. Fix $\rho \in (0,1)$. Let $z_{1},\ldots ,z_{m}\in \mathbb {R}^{n+1}$ be the vertices of a regular simplex in $\mathbb {R}^{n+1}$ centred at the origin. Then $\exists \ w\in \mathbb {R}^{n+1}$ such that, for all $1\leq i\leq m$,

$$ \begin{align*}\Omega_{i}=w+\{x\in\mathbb{R}^{n+1}\colon\langle x,z_{i}\rangle=\max_{1\leq j\leq m}\langle x,z_{j}\rangle\}.\end{align*} $$

It is known that Conjecture 1.6 is false when $(a_{1},\ldots ,a_{m})\neq (1/m,\ldots ,1/m)$ [Reference Heilman, Mossel and NeemanHMN16]. In the remaining case that $a_{i}=1/m$ for all $1\leq i\leq m$, it is assumed that $w=0$ in Conjecture 1.6.

For expositional simplicity, we separately address the case $\rho <0$ of Conjecture 1.6 in Section 7.

1.3 Plurality Is Stablest Conjecture

As previously mentioned, the standard simplex conjecture [Reference Isaksson and MosselIM12] stated in Conjecture 1.6 is essentially equivalent to the plurality is stablest conjecture from Conjecture 1.2. After making several definitions, we state a formal version of Conjecture 1.2 as Conjecture 1.7.

If $g\colon \{1,\ldots ,m\}^{n}\to \mathbb {R}$ and $1\leq i\leq n$, we denote

$$ \begin{align*}\mathbb{E}(g)\colon= m^{-n}\sum_{\omega\in\{1,\ldots,m\}^{n}} g(\omega)\end{align*} $$

$$ \begin{align*}\mathbb{E}_{i}(g)(\omega_{1},\ldots,\omega_{i-1},\omega_{i+1},\ldots,\omega_{n})\colon= m^{-1}\sum_{\omega_{i}\in\{1,\ldots,m\}} g(\omega_{1},\ldots,\omega_{n})\end{align*} $$

$$ \begin{align*}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall\,(\omega_{1},\ldots,\omega_{i-1},\omega_{i+1},\ldots,\omega_{n})\in\{1,\ldots,m\}^{n}.\end{align*} $$

Define also the ith influence of g – that is, the influence of the $i^{th}$ voter of g – as

(5)$$ \begin{align} \mathrm{Inf}_{i}(g)\colon= \mathbb{E} [(g-\mathbb{E}_{i}g)^{2}]. \end{align} $$

Let

(6)$$ \begin{align} \Delta_{m}\colon=\{(y_{1},\ldots,y_{m})\in\mathbb{R}^{m}\colon y_{1}+\cdots+y_{m}=1,\,\forall\,1\leq i\leq m,\,y_{i}\geq0\}. \end{align} $$

If $f\colon \{1,\ldots ,m\}^{n}\to \Delta _{m}$, we denote the coordinates of f as $f=(f_{1},\ldots ,f_{m})$. For any $\omega \in \mathbb {Z}^{n}$, we denote $\left \|\omega \right \|_{0}$ as the number of nonzero coordinates of $\omega $. The noise stability of $g\colon \{1,\ldots ,m\}^{n}\to \mathbb {R}$ with parameter $\rho \in (-1,1)$ is

$$ \begin{align*} S_{\rho} g \colon =\,& m^{-n}\sum_{\omega\in\{1,\ldots,m\}^{n}} g(\omega)\mathbb{E}_{\rho} g(\delta)\\ =\,& m^{-n}\sum_{\omega\in\{1,\ldots,m\}^{n}} g(\omega)\sum_{\sigma\in\{1,\ldots,m\}^{n}}\left(\frac{1-(m-1)\rho}{m}\right)^{n-\left\|\sigma-\omega\right\|_{0}} \left(\frac{1-\rho}{m}\right)^{\left\|\sigma-\omega\right\|_{0}} g(\sigma). \end{align*} $$

Equivalently, conditional on $\omega $, $\mathbb {E}_{\rho }g(\delta )$ is defined so that for all $1\leq i\leq n$, $\delta _{i}=\omega _{i}$ with probability $\frac {1-(m-1)\rho }{m}$, and $\delta _{i}$ is equal to any of the other $(m-1)$ elements of $\{1,\ldots ,m\}$ each with probability $\frac {1-\rho }{m}$, so that $\delta _{1},\ldots ,\delta _{n}$ are independent.

The noise stability of $f\colon \{1,\ldots ,m\}^{n}\to \Delta _{m}$ with parameter $\rho \in (-1,1)$ is

$$ \begin{align*}S_{\rho}f\colon=\sum_{i=1}^{m}S_{\rho}f_{i}.\end{align*} $$

Let $m\geq 2$, $k\geq 3$. For each $j\in \{1,\ldots ,m\}$, let $e_{j}=(0,\ldots ,0,1,0,\ldots ,0)\in \mathbb {R}^{m}$ be the jth unit coordinate vector. Define the plurality function $\mathrm {PLUR}_{m,n}\colon \{1,\ldots ,m\}^{n}\to \Delta _{m}$ for m candidates and n voters such that for all $\omega \in \{1,\ldots ,m\}^{n}$.

$$ \begin{align*}\mathrm{PLUR}_{m,n}(\omega) \colon=\begin{cases} e_{j}&,\mbox{if }\left|\{i\in\{1,\ldots,m\}\colon\omega_{i}=j\}\right|>\left|\{i\in\{1,\ldots,m\}\colon\omega_{i}=r\}\right|,\\ &\qquad\qquad\qquad\qquad\forall\,r\in\{1,\ldots,m\}\setminus\{j\}\\ \frac{1}{m}\sum_{i=1}^{m}e_{i}&,\mbox{otherwise}. \end{cases} \end{align*} $$

We can now state the more formal version of Conjecture 1.2.

Conjecture 1.7 Plurality Is Stablest, Discrete Version

For any $m\geq 2$, $\rho \in [0,1]$, $\varepsilon>0$, there exists $\tau>0$ such that if $f\colon \{1,\ldots ,m\}^{n}\to \Delta _{m}$ satisfies $\mathrm {Inf}_{i}(f_{j})\leq \tau $ for all $1\leq i\leq n$ and for all $1\leq j\leq m$, and if $\mathbb {E} f=\frac {1}{m}\sum _{i=1}^{m}e_{i}$, then

$$ \begin{align*}S_{\rho}f\leq \lim_{n\to\infty}S_{\rho}\mathrm{PLUR}_{m,n}+\varepsilon. \end{align*} $$

The main result of the present article (stated in Theorem 1.10) is $\exists \ \rho _{0}>0$ such that Conjecture 1.7 is true for $m=3$ for all $0<\rho <\rho _{0}$, for all $n\geq 1$. The only previously known case of Conjecture 1.7 was the following.

For an even more general version of Theorem 1.8, see [Reference Mossel, O’Donnell and OleszkiewiczMOO10, Theorem 4.4]. In particular, the assumption on $\mathbb {E} f$ can be removed, though we know that this cannot be done for $m\geq 3$ [Reference Heilman, Mossel and NeemanHMN16].

1.4 Our Contribution

The main structure theorem below implies that sets optimising noise stability in Problem 1.5 are inherently low-dimensional. Though this statement might seem intuitively true, because many inequalities involving the Gaussian measure have low-dimensional optimisers, this statement has not been proven before. For example, Theorem 1.9 was listed as an open question in [Reference De, Mossel and NeemanDMN17, Reference De, Mossel and NeemanDMN18] and [Reference Ghazi, Kamath and RaghavendraGKR18]. Indeed, the lack of Theorem 1.9 has been one main obstruction to a solution of Conjectures 1.5 and 1.7.

Theorem 1.9 Main Structure Theorem/Dimension Reduction

Fix $\rho \in (0,1)$. Let $m\geq 2$ with $m\leq n+2$. Let $\Omega _{1},\ldots \Omega _{m}\subseteq \mathbb {R}^{n+1}$ maximise Problem 1.5. Then, after rotating the sets $\Omega _{1},\ldots \Omega _{m}$ and applying Lebesgue measure zero changes to these sets, there exist measurable sets $\Omega _{1}',\ldots \Omega _{m}'\subseteq \mathbb {R}^{m-1}$ such that

$$ \begin{align*}\Omega_{i}=\Omega_{i}'\times\mathbb{R}^{n-m+2},\qquad\forall\, 1\leq i\leq m.\end{align*} $$

In the case $m=2$, Theorem 1.9 is (almost) a variational proof of Borell’s inequality, because it reduces Problem 1.5 to a 1-dimensional problem.

In the case $m=3$, Theorem 1.9 says that Conjecture 1.6 for arbitrary $n+1$ reduces to the case $n+1=2$, which was solved for small $\rho>0$ in [Reference HeilmanHei14]. That is, Theorem 1.9 and the main result of [Reference HeilmanHei14] imply the following.

In [Reference HeilmanHei14] it is noted that $\rho _{0}=e^{-20\cdot 3^{10^{14}}}$ suffices in Theorem 1.10.

We can also prove a version of Theorem 1.9 when $\rho <0$. See Theorem 7.9 and the discussion in Section 7. One difficulty in proving Theorem 1.9 directly for $\rho <0$ is that it is not a priori obvious that a minimiser of Problem 1.5 exists in that case.

1.5 Noninteractive Simulation of Correlated Distributions

As mentioned above, Theorem 1.9 answers a question in [Reference De, Mossel and NeemanDMN17, Reference De, Mossel and NeemanDMN18] and [Reference Ghazi, Kamath and RaghavendraGKR18]. Their interest in Theorem 1.9 stems from the following problem. Let $(X,Y)\in \mathbb {R}^{n}$ be a random vector. Let $(X_{1},Y_{1}),(X_{2},Y_{2}),\ldots $ be independent and identically distributed copies of $(X,Y)$. Suppose there are two players A and B. Player A has access to $X_{1},X_{2},\ldots $ and player B has access to $Y_{1},Y_{2},\ldots $. Without communication, what joint distributions can players A and B jointly simulate? For details on the relation of this problem to Theorem 1.9, see [Reference De, Mossel and NeemanDMN17, Reference De, Mossel and NeemanDMN18] and [Reference Ghazi, Kamath and RaghavendraGKR18].

1.6 Outline of the Proof of the Structure Theorem

In this section we outline the proof of Theorem 1.9 in the case $m=2$. The proof loosely follows that of a corresponding statement [Reference McGonagle and RossMR15, Reference Barchiesi, Brancolini and JulinBBJ17] for the Gaussian surface area (which was then adapted to multiple sets in [Reference Milman and NeemanMN18a, Reference Milman and NeemanMN18b, Reference HeilmanHei18]), with a few key differences. For didactic purposes, we will postpone a discussion of technical difficulties (such as existence and regularity of a maximiser) to Subsection 2.1.

Fix $0<a<1$. Suppose there exists $\Omega ,\Omega ^{c}\subseteq \mathbb {R}^{n+1}$ are measurable sets maximizing

$$ \begin{align*}\int_{\mathbb{R}^{n+1}}1_{\Omega}(x)T_{\rho}1_{\Omega}(x)\gamma_{n+1}(x)dx,\end{align*} $$

subject to the constraint $\gamma _{n+1}(\Omega )=a$. A first variation argument (Lemma 3.1) implies that $\Sigma \colon =\partial \Omega $ is a level set of the Ornstein–Uhlenbeck operator applied to $1_{\Omega }$. That is, there exists $c\in \mathbb {R}$ such that

(7)$$ \begin{align} \Sigma=\{x\in\mathbb{R}^{n+1}\colon T_{\rho}1_{\Omega}(x)=c\}. \end{align} $$

Because $\Sigma $ is a level set, a vector perpendicular to the level set is also perpendicular to $\Sigma $. Denoting $N(x)\in \mathbb {R}^{n+1}$ as the unit length exterior pointing normal vector to $x\in \partial \Omega $, (7) implies that

(8)$$ \begin{align} \overline{\nabla}T_{\rho}1_{\Omega}(x)= -N(x)\|\overline{\nabla}T_{\rho}1_{\Omega}(x)\|. \end{align} $$

(It is not obvious that there must be a negative sign here, but it follows from examining the second variation.) We now observe how the noise stability of $\Omega $ changes as the set is translated infinitesimally. Fix $v\in \mathbb {R}^{n+1}$ and consider the variation of $\Omega $ induced by the constant vector field v. That is, let $\Psi \colon \mathbb {R}^{n+1}\times (-1,1)\to \mathbb {R}^{n+1}$ such that $\Psi (x,0)=x$ and such that $\frac {\mathrm {d}}{\mathrm {d} s}|_{s=0}\Psi (x,s)=v$ for all $x\in \mathbb {R}^{n+1},s\in (-1,1)$. For any $s\in (-1,1)$, let $\Omega ^{(s)}=\Psi (\Omega ,s)$. Note that $\Omega ^{(0)}=\Omega $. Denote $f(x)\colon = \langle v,N(x)\rangle $ for all $x\in \Sigma $. Then define

$$ \begin{align*}S(f)(x)\colon= (1-\rho^{2})^{-(n+1)/2}(2\pi)^{-(n+1)/2}\int_{\Sigma}f(y)e^{-\frac{\left\|y-\rho x\right\|^{2}}{2(1-\rho^{2})}}\,\mathrm{d} y,\qquad\forall\,x\in\Sigma. \end{align*} $$

A second variation argument (Lemma 4.5) implies that, if f is Gaussian volume preserving – that is, $\int _{\Sigma }f(x)\gamma _{n+1}(x)\,\mathrm {d} x=0$ – then

(9)$$ \begin{align} \begin{aligned} &\frac{1}{2}\frac{\mathrm{d}^{2}}{\mathrm{d} s^{2}}\Big|_{s=0}\int_{\mathbb{R}^{n+1}}1_{\Omega^{(s)}}(x)T_{\rho}1_{\Omega^{(s)}}(x)\gamma_{n+1}(x)\,\mathrm{d} x\\ &\qquad\qquad\qquad\qquad=\int_{\Sigma}\Big(S(f)(x)-\|\overline{\nabla}T_{\rho}1_{\Omega}(x)\|f(x)\Big)f(x)\gamma_{n+1}(x)\,\mathrm{d} x. \end{aligned} \end{align} $$

Somewhat unexpected, the function $f(x)=\langle v,N(x)\rangle $ is almost an eigenfunction of the operator S (by Lemma 5.1), in the sense that

(10)$$ \begin{align} S(f)(x)=\frac{1}{\rho}f(x)\|\overline{\nabla} T_{\rho}1_{\Omega}(x)\|,\qquad\forall\,x\in\Sigma. \end{align} $$

Equation (10) is the key fact used in the proof of the main theorem, Theorem 1.9. Equation (10) follows from (8) and the divergence theorem (see Lemma 5.1 for a proof of (10).) Plugging (10) into (9),

(11)$$ \begin{align} \begin{aligned} \int_{\Sigma}\langle v,N(x)\rangle\gamma_{n+1}(x)\,\mathrm{d} x=0\quad\Longrightarrow\quad &\frac{1}{2}\frac{\mathrm{d}^{2}}{\mathrm{d} s^{2}}\Big|_{s=0}\int_{\mathbb{R}^{n+1}}1_{\Omega^{(s)}}(x)T_{\rho}1_{\Omega^{(s)}}(x)\gamma_{n+1}(x)\,\mathrm{d} x\\ &\qquad=\Big(\frac{1}{\rho}-1\Big)\int_{\Sigma}\langle v,N(x)\rangle^{2}\left\|\overline{\nabla}T_{\rho}1_{\Omega}(x)\right\|\gamma_{n+1}(x)\,\mathrm{d} x. \end{aligned} \end{align} $$

The set

$$ \begin{align*} V\colon=\Big\{v\in\mathbb{R}^{n+1}\colon \int_{\Sigma}\langle v,N(x)\rangle\gamma_{n+1}(x)\,\mathrm{d} x=0\Big\} \end{align*} $$

has dimension at least n, by the rank-nullity theorem. Because $\Omega $ maximises noise stability, the quantity on the right of (11) must be nonpositive for all $v\in V$, implying that $f=0$ on $\Sigma $ (except possibly on a set of measure zero on $\Sigma $. One can show that $\left \|\overline {\nabla }T_{\rho }1_{\Omega }(x)\right \|>0$ for all $x\in \Sigma $. See Lemma 4.8.) That is, for all $v\in V$, $\langle v,N(x)\rangle =0$ for all $x\in \Sigma $ (except possibly on a set of measure zero on $\Sigma $). Because V has dimension at least n, there exists a measurable discrete set $\Omega '\subseteq \mathbb {R}$ such that $\Omega =\Omega '\times \mathbb {R}^{n}$ after rotating $\Omega $, concluding the proof of Theorem 1.9 in the case $m=2$.

Theorem 1.9 follows from the realization that all of the above steps still hold for arbitrary m in Conjecture 1.5. In particular, the key lemma (10) still holds. See Lemmas 5.1 and 5.4.

Remark 1.11. In the case that we replace the Gaussian noise stability of $\Omega $ with the Euclidean heat content

$$ \begin{align*}\int_{\mathbb{R}^{n+1}}1_{\Omega}(x)P_{t}1_{\Omega}(x)\,\mathrm{d} x,\qquad\forall\,t>0 \end{align*} $$

$$ \begin{align*}P_{t}f(x) \colon=\int_{\mathbb{R}^{n+1}}f(x+y\sqrt{t})\gamma_{n+1}(y)\,\mathrm{d} y, \qquad\forall x\in\mathbb{R}^{n+1},\qquad\forall\,f\colon\mathbb{R}^{n+1}\to[0,1],\end{align*} $$

the corresponding operator $\overline {S}$ from the second variation of the Euclidean heat content satisfies

$$ \begin{align*}\overline{S}(f)(x)\colon= t^{(n+1)/2}(2\pi)^{-(n+1)/2}\int_{\Sigma}f(y)e^{-\frac{\left\|y- x\right\|^{2}}{2t}}\,\mathrm{d} y,\qquad\forall\,x\in\Sigma, \end{align*} $$

and then the analogue of (9) for $f(x)\colon =\langle v,N(x)\rangle $ is

$$ \begin{align*}\overline{S}(f)(x)=f(x)\|\overline{\nabla} P_{t}1_{\Omega}(x)\|,\qquad\forall\,x\in\Sigma,\end{align*} $$

so that the second variation corresponding to $f=\langle v,N\rangle $ is automatically zero. This fact is expected, because a translation does not change the Euclidean heat content. However, this example demonstrates that the key property of the above proof is exactly (10). More specifically, f is an ‘almost eigenfunction’ of S with ‘eigenvalue’ $1/\rho $ that is larger than $1$. It seems plausible that other semigroups could also satisfy an identity such as (10), because (10) seems related to hypercontractivity. We leave this open for further research.