Euclidean Partitions Optimizing Noise Stability

The Standard Simplex Conjecture of Isaksson and Mossel asks for the partition $\{A_{i}\}_{i=1}^{k}$ of $\mathbb{R}^{n}$ into $k\leq n+1$ pieces of equal Gaussian measure of optimal noise stability. That is, for $\rho>0$, we maximize $$ \sum_{i=1}^{k}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}1_{A_{i}}(x)1_{A_{i}}(x\rho+y\sqrt{1-\rho^{2}}) e^{-(x_{1}^{2}+\cdots+x_{n}^{2})/2}e^{-(y_{1}^{2}+\cdots+y_{n}^{2})/2}dxdy. $$ Isaksson and Mossel guessed the best partition for this problem and proved some applications of their conjecture. For example, the Standard Simplex Conjecture implies the Plurality is Stablest Conjecture. For $k=3,n\geq2$ and $0<\rho<\rho_{0}(k,n)$, we prove the Standard Simplex Conjecture. The full conjecture has applications to theoretical computer science, and to geometric multi-bubble problems (after Isaksson and Mossel).


Introduction
The Standard Simplex Conjecture [12] asks for the partition {A i } k i=1 of R n into k ≤ n + 1 sets of equal Gaussian measure of optimal noise stability. This Conjecture generalizes a seminal result of Borell, [3,19], which corresponds to the k = 2 case of the Standard Simplex Conjecture. Borell's result says that the two disjoint regions of fixed Gaussian measures 0 < a < 1 and 1 − a and of optimal noise stability must be separated by a hyperplane. Since two disjoint sets of total Gaussian measure 1 can be described by a single set and its complement, Borell's result can be stated as follows. Let A ⊆ R n have Gaussian measure 0 < a < 1 and let ρ ∈ (0, 1). Then the following quantity, which is referred to as the noise stability of A, is maximized when A is a half-space. (1) When we say that A is a half-space, we mean that A is the set of points lying on one side of a hyperplane. If ρ ∈ (−1, 0), then the noise stability (1) of A is minimized among all sets of Gaussian measure a, when A is a half-space. We can rewrite (1) probabilistically as follows. Let X = (X 1 , . . . , X n ), Y = (Y 1 , . . . , Y n ) ∈ R n be two standard Gaussian random vectors such that E(X i Y j ) = ρ · 1 (i=j) . Then the noise stability (1) of A is equal to P((X, Y ) ∈ A × A). For modern proofs of Borell's theorem with additional stability statements, see [18,7]. In the present work, we prove a specific case of the Standard Simplex Conjecture for k = 3, when 0 < ρ < ρ 0 (n). Already for the case k = 3, the methods used in the case k = 2 do not seem to apply, so new techniques are required to treat the case k = 3. We first discuss The problem of minimizing Gaussian perimeter arises as an endpoint case of the Standard Simplex Conjecture. The Standard Simplex Conjecture is a statement involving a sum of terms of the form (1), and the Gaussian perimeter is recovered by letting ρ → 1 − .
Definition 1.1. Let A 1 , . . . , A k ⊆ R n be measurable, k ≤ n + 1. We say that be the vertices of a regular simplex centered at the origin of R n . For each i ∈ {1, . . . , k}, define A i := {x ∈ R n : x, z i = max j∈{1,...,k} x, z j }, the Voronoi region of z i . We call a regular simplicial conical partition.
The following theorem is our main result.
1.1. MAX-k-CUT and the Unique Games Conjecture. We now rigorously describe the complexity theoretic notions referenced above. Definition 1.3 (MAX-k-CUT). Let k, n ∈ N, k ≥ 2. We define the weighted MAX-k-CUT problem. We are given a symmetric matrix {a ij } n i,j=1 with a ij ≥ 0 for all i, j ∈ {1, . . . , n}. The goal of the MAX-k-CUT problem is to find the following quantity: Definition 1.4 (Γ-MAX-2LIN(k)). Let k ∈ N, k ≥ 2. We define the Γ-MAX-2LIN(k) problem. In this problem, we are given m ∈ N and 2m variables x i ∈ Z/kZ, i ∈ {1, . . . , 2m}.
If (4) were equal to m, then we could find (x 1 , . . . , x 2m ) achieving the maximum in (4) by linear algebra. One can therefore interpret the Unique Games Conjecture as an assertion that approximate linear algebra is hard. Theorem 1.6. (Optimal Approximation for MAX-k-CUT, [12][Theorem 1.13], [8]). Let k ∈ N, k ≥ 2. Let {A i } k i=1 ⊆ R k−1 be a regular simplicial conical partition. Define .
Assume Conjecture 1 and the Unique Games Conjecture. Then, for any ε > 0, there exists a polynomial time algorithm that approximates MAX-k-CUT within a multiplicative factor α k −ε, and it is NP-hard to approximate MAX-k-CUT within a multiplicative factor of α k +ε.
So, for small ρ > 0, a partition maximizing (d/dρ)J is close to a regular simplicial conical partition. The structure of the operator T ρ then permits the exploitation of a feedback loop. This feedback loop says: if our partition maximizes (d/dρ)J for small ρ > 0, and if this partition is close to a regular simplicial conical partition, then this partition is even closer to a regular simplicial conical partition. This feedback loop is investigated in Section 5, especially in the crucial Lemma 6.1. A similar feedback loop was already apparent in [14][ Lemma 3.3]. The full argument of Theorem 1.2 is then assembled in Section 7. By using this feedback loop, we show in Theorem 7.1 that a regular simplicial conical partition maximizes (d/dρ)J for small ρ > 0, k = 3, n ≥ 2. Then, the Fundamental Theorem of Calculus allows us to relate (d/dρ)J to J, therefore completing the proof of the main theorem, Theorem 1.2.
Since Lemma 6.1 is rather lengthy and crucial to this investigation, we will further describe the idea behind it. If we know that our partition maximizes (d/dρ)J, and if we also know that this partition is close to a regular simplicial conical partition, then the first variation should immediately tell us that our partition is actually a regular simplicial conical partition. Unfortunately, this intuition does not translate into a proof. The main technical problem is that the sets we are dealing with are unbounded, and we need to know precise information about the Ornstein-Uhlenbeck operator applied to these sets, for points that are very far from the origin. Since the Gaussian measure decays exponentially away from the origin, this means that it becomes hard to say something precise about the points in these sets that are very far from the origin. So, we require very precise estimates of the Ornstein-Uhlenbeck operator, and the errors that it accrues when we evaluate it far from the origin. These estimates are performed in Lemmas 5.2 and 5.3. Unfortunately, to use these estimates effectively, we need to slowly enlarge the regions where we use these estimates. The details of enlarging these regions becomes surprisingly complicated, occupying the seven steps of Lemma 6.1.
In Section 7, we also show the surprising fact that our strategy fails for small negative correlation. That is, for small ρ < 0, (d/dρ)J is not maximized by the regular simplicial conical partition. This result does not confirm or deny Conjecture 1 for ρ < 0. However, one may interpret from this result that the case of Conjecture 1 for ρ < 0 could be more difficult than the case ρ > 0.
We should also emphasize the lack of symmetrization in the proof of Theorem 1.2. Symmetrization is one of a few general strategies that solves many optimization problems. In our context, symmetrization would appear as follows. Recall the definition of J from (3).
In the proof of the main theorem, it is tempting to use this symmetrization paradigm. The works [3], [19] and [12] use Gaussian symmetrization in a crucial way. However, we find this approach to be less natural for Conjecture 1, so we do not explicitly use symmetrization. Nevertheless, symmetry does play a crucial role in our proof, especially in the estimates of Section 4. It should also be noted that the works [14,16] do not explicitly use symmetrization, and this lack of symmetrization is one of their novel aspects.

7
A well-known calculation shows the following equality, which we prove in the Appendix, Section 9.
We say that A ⊆ R n is a cone if A is measurable and ∀ t > 0, tA = A.
is a partition of R n together with a simplex S ⊆ R k−1 with 0 ≤ k − 1 ≤ n and a rotation σ of R n such that 0 ∈ S and such that each facet F i of σ(S × R n−k+1 ) generates a partition element, i.e.
⊆ R n be nonzero vectors that do not all lie in a (k − 1)-dimensional hyperplane. Define a partition such that, for i ∈ {1, . . . , k}, A i := {x ∈ R n : x, z i = max j=1,...,k x, z j }. Such a partition is called the simplicial conical partition induced by , then we say the partition is a balanced conical partition. If {z i } k i=1 ⊆ R n are the vertices of a (k − 1)-dimensional regular simplex in R n centered at the origin, then the partition induced by {z i } k i=1 is called a regular simplicial conical partition. Let f ∈ L 2 (γ n ). By Plancherel and (7) Taking the derivative d/dρ of (12) at ρ = 0, we get a quantity studied in [14,16].

Noise Stability for Zero Correlation
This section concerns noise stability at the endpoint ρ = 0. Specifically, we will investigate the quantity (14), which has already been studied in [14,16]. Using our understanding of (14), we will then be able to analyze the left side of (13) when ρ is small, using the equality (13). Before beginning our discussion, we first need to consider partitions of R n within the convex set defined in (15). Definition 2.3 provides the metric allowing an assertion that two partitions are close to each other, and Definition 2.4 allows us to discuss the Gaussian measure restricted to hypersurfaces.
The next two lemmas are derived from [14]. Lemma 2.6 is a quantitative variant of Lemma 2.5, and it will be further improved in Lemma 2.8 below. In particular, Lemma 2.6 says that, if the first variation condition for achieving the optimum value of (16) is nearly satisfied, then the partition is close to being simplicial.
Lemma 2.6. Let n ≥ 2 and let So, to complete the proof, it suffices to show that the second case does not occur. We find a contradiction by assuming that the second case occurs.
We require the ensuing explicit calculation from [14] in Lemma 2.8 below. This calculation is reduced to a computation of Lagrange Multipliers in [14,Corollary 3.4]. For any The following Lemma is a quantitative improvement of Lemmas 2.5 and 2.6. Combining Lemma 2.8 with (14) will show that an optimizer of (d be a regular simplicial conical partition of R n . Assume that ε < 1/100 and Then Proof. Assume that (17) Rewriting this inequality using the definition of ψ 0 , Since {A ′′ p } 3 p=1 is a partition of R n with at most two nonempty elements, Lemma 2.7 says Combining (19) and (20) contradicts (17). Therefore, z i , z j < 0 for all i, j ∈ {1, 2, 3}. We now claim that, for each pair i, j ∈ {1, 2, 3} with i = j, we have max p∈{i,j} We again argue by contradiction. Suppose there exist i, j ∈ {1, 2, 3} with i = j and max p∈{i,j} z p with equality if and only if 1 Ap is a half-space whose boundary contains the origin of R n . This follows immediately from rearrangement. Observe, if z p = 0, . . , f k ) = 9/(8π), using Lemma 2.7. We conclude that (21) holds.

The First Variation
Recall ( Also, for each i ∈ {1, . . . , k}, the following containment holds, less sets of γ n measure zero: Proof. We show that (3) is maximized over ∆ k (γ n ), which contains the set of partitions of R n . Note that ∆ k (γ n ) ⊆ H is norm closed, convex, and norm bounded. Therefore, ∆ k (γ n ) is weakly closed. Also, ∆ k (γ n ) is weakly compact by the Banach-Alaoglu Theorem. Using By (12), ψ ρ is an exponentially decaying sum of uniformly bounded weakly continuous functions. Therefore, ψ ρ is weakly continuous on the weakly compact set ∆ k (γ n ). So there Since ψ ρ is convex on ∆ k (γ n ), ψ ρ achieves its maximum at an extreme point of ∆ k (γ n ). Therefore, there exists a partition Therefore, there exists a ball B(y, r), r > 0 such that γ n (B(y, r) ∩ A j ) > 0 and such that sup x∈B(y,r) However, (31) But (31) contradicts the maximality of (1 A 1 , . . . , 1 A k ) on ∆ k (γ n ), so (28) holds.

Perturbative Estimates
Recalling (29), the following estimates allow us to relate ψ ρ to ψ 0 for small ρ > 0, for simplicial conical partitions. In particular, we make a close examination of the two quantities of (10). Since lemma 4.2 gives precise estimates of the two quantities of (10), combining Lemma 4.2 with (28) gives precise geometric information about a partition {A i } k i=1 ⊆ R n optimizing noise stability. In particular, to see one way that we will apply Lemma 4.2, see (121) below. However, note that (121) below does not give sufficiently precise information to identify the sets optimizing noise stability. So, the real need for Lemma 4.2 will occur in the proof of the Main Lemma 6.1, where the precise estimate (51) is used.
Proof. The assertion follows by standard equalities for the moments of a Gaussian random variable. Let α > 0. Define f (α) by the formula By changing variables, f (α) = α −n/2 R n 1 A (y)dγ n (y). So, Note that span{z i , z j } = span{e 1 , e 2 }. Let n j ∈ R n be the interior unit normal of B j so that n j is normal to the face (∂B j ) \ (∂B i ), and let n i ∈ R n be the interior unit normal of B i so that n i is normal to the face (∂B i ) \ (∂B j ).
Proof of (i). Below, we use differentiation in the distributional sense. Let Here we used Let x with x ∈ B i and x, (−n j ) ≥ 0. Then (36) immediately proves (32). Define By (37), w is in the convex hull of e 1 and n i . In particular, x, w ≥ 0, since x ∈ B i . Combining x, w ≥ 0 with (36) proves (33).

Iterative Estimates
The following estimates control the errors that appear in the proof of Theorem 1.2. Being rather technical in nature, this section could be skipped on a first reading.
Also, for m = 0 we have So, combining the above estimates with (38), Therefore, for ℓ = (ℓ 1 , . . . , ℓ n ) ∈ N n , The following Lemma uses standard tail bounds for a Gaussian random variable. We therefore omit the proof.
The following Lemma says, if R n xf (x)dγ n (x) is parallel to the x 1 -axis, then (d/dρ)T ρ f (x) should be bounded by a constant multiplied by |x 1 | + O(ρ). The precise error term (39) will be needed in Lemma 6.1 to determine the size of (d/dρ)T ρ (1 A i − 1 A j ). The error term (39) will be estimated by Lemma 5.2, and the resulting estimate will be introduced into (28).
So, using R n y 2 f (y)dγ n (y) = 0 and the Fundamental Theorem of Calculus, By integrating by parts again, note that Applying the Fundamental Theorem of Calculus to (41) and then using (40), By integrating by parts as before, Combining (10), (42) and (43), We then deduce (39) from (44).

The Main Lemma
Lemma 6.1 below represents the main tool in the proof of the main theorem. As depicted in Figure 1, Lemma 6.1 says that, if an optimal partition is close to being simplicial conical, then it is actually much closer to being simplicial conical. So, this Lemma can be understood as a feedback loop, or as a contractive mapping type of argument. We first give an intuitive sketch of the proof of the Lemma. Let ρ > 0. We begin with a partition {A p } 3 p=1 ⊆ R n maximizing noise stability (3). We assume that there are disjoint sets {D p } 3 p=1 that resemble a simplicial conical partition, as in the left side of Figure 1. We also assume that A p ⊇ D p for all p = 1, 2, 3. We then find a sequence of sets {D p,1 } 3 p=1 , {D p,2 } 3 p=1 , . . . {D p,R } 3 p=1 such that D p,r ⊆ D p,r+1 for all 1 ≤ p ≤ 3, for all r ≥ 1. This sequence of sets is chosen so that the following implication can be proven: In order to prove (45), we need to show: if A p ⊇ D p,r , then we can get sufficiently strong estimates on LT ρ 1 D p,r+1 such that (28) can be verified on A p for each p = 1, 2, 3. For example, in Step 1 of the proof of Lemma 6.1, the estimate (54) eventually implies (58). And (58) says that A p must contain more points than the initial information that we assumed in (48). Finally, we need to choose our sets {D p,r } 3 p=1 appropriately so that, after finitely many implications of the form (45), we eventually get the conclusion (49). That is, the three sets {D p,R } 3 p=1 resemble the right side of Figure 1, and A p ⊇ D p,R for each p = 1, 2, 3. Thus concludes our description of the main strategy of the proof. Within the proof itself, the sets {D p,1 } 3 p=1 , {D p,2 } 3 p=1 , . . . will not be explicitly defined. However, portions of these sets will be defined at the end of every Step of the proof. In particular, examine the sets defined by the following sequence of assertions: Unfortunately, there are many technical obstacles that stand in the way of bringing this strategy to fruition. The first minor issue is that we cannot control small rotations of our sets. At every step of the proof, we therefore need to redefine our simplicial sets B i , B j to account for these small rotations. However, the main technical issue is that it is not at all obvious how to choose the sets {D p,r } 3 p=1 for r = 1, 2, 3, . . . such that (45) can be proven for each r = 1, 2, 3, . . .. Moreover, the simplest choice of these sets, namely dilations of the sets depicted in Figure 1, do not produce satisfactory estimates.
Ultimately, the sequence of sets defined by (58), (63), (68), (78), (88), (95), (101) succeeds in proving the sequence of implications (45) for r = 1, 2, 3, . . .. Lemma 5.3 allows us to control the errors from our estimates, and we then make around seven modifications of the same error estimate within Lemma 6.1. This error estimate allows us to apply Lemma 4.2, so that we can improve our knowledge of the optimal partition {A i } k i=1 via (28). It would be preferable to write Lemma 6.1 as seven applications of a single Lemma, however the statement of such a Lemma would perhaps be so long and convoluted that its application would become opaque. We therefore use the longer presentation below in the hope of providing greater clarity. Finally, in the statement of Lemma 6.1 below, note that the plane Π exists independently of i, j ∈ {1, . . . , k}.
The estimate (68) now has a cascading effect on the estimates below. From (68), We already verified the case M = 0, K = 1. We assume that, for 0 ≤ m < M, Assume also that, for M ≤ m ≤ √ K − 1 and K ≥ 1, We will conclude that (70) holds for m = M, i.e.
We perform another induction, though this time we hold K fixed and use the additional ingredient (78). Let M, R ∈ N with 0 ≤ M ≤ √ K, R ≥ 0 such that ρ .9(K+R) > η 1/5 . We will induct on M and R. We assume that, for 0 ≤ m < M, We know that the case R = 0, 0 ≤ M ≤ √ K of (79) holds by (72). We therefore assume that R ≥ 1. Assume also that, for M ≤ m ≤ √ K, We will conclude that (79) holds for m = M, i.e.
In the latter case, (88) follows, and in the former cases, (87) implies In all cases, (88) holds. We can finally use (88) to conclude the proof.

Proof of the Main Theorem
We now combine the Lemmas of the previous sections, as described in Section 1. The main effort involves verifying the assumption of the Main Lemma 6.1. Once this is done, Lemma 6.1 can be iterated infinitely many times to complete the proof. Theorem 7.1. Fix k = 3, n ≥ 2. Define ∆ k (γ n ) as in Definition 2.1 and define ψ ρ as in (29). Let {C i } k i=1 ⊆ R n be a regular simplicial conical partition. Then there exists ρ 0 = ρ 0 (n, k) > 0 such that, for all ρ ∈ (0, ρ 0 ), (1 C 1 , . . . , 1 C k ) uniquely achieves the following supremum, up to rotation Proof. Within the proof, we will assert that ρ > 0 satisfies several upper bounds, and then at the end of the proof, we will define ρ 0 as the minimum of these upper bounds. By Lemma 3.1, let {A i } k i=1 be a partition of R n such that By (12), write Step 1. The partition {A i } k i=1 is close to being simplicial. For i ∈ {1, . . . , k}, let z i := A i xdγ n (x) ∈ R n . Subtracting the |ℓ| = 1 term from both sides of (103), treating the remaining terms as error terms, and using that 1 A i L 2 (γn) ≤ 1 for all i = 1, . . . , k, Therefore, Step 2. Applying a small rotation.
Let {B ′ p } k p=1 be a regular simplicial conical partition such that such that for fixed i = j, i, j ∈ {1, . . . , k} and for some λ ′ ∈ R, and such that Such {B ′ p } k p=1 exists by (109), letting ρ > 0 such that ρ < (10000k) −8 , so that So, by the triangle inequality applied to (109), and (110), Specifically, we first apply a rotation to {B ′′ p } k p=1 such that (112) holds. Then, by (110), we then apply another rotation that fixes the x 1 axis, so that (113) holds. By (109), (110), (114) and (115), each of these two rotations can be chosen so that a given unit vector is moved in R n a distance not more than 12(6kρ) 1/8 . And since we are rotating three polygonal cones with two facets each, (111) holds.
Using (111) and the triangle inequality, Also, using that c ℓ = 0 for |ℓ| = 1, ℓ 1 = 0, (112) implies that B ′ i ∩ B ′ j ⊆ {x ∈ R n : x 1 = 0}, and we may assume that B ′ i ⊆ {x ∈ R n : x 1 ≥ 0}. Let n ′ i ∈ R n denote the interior unit normal of B ′ i such that n ′ i is normal to (∂B ′ i ) \ B ′ j , and let n ′ j ∈ R n denote the interior unit normal of B ′ j such that n ′ j is normal to (∂B ′ j ) \ B ′ i . Then, define B i , B j such that Step 3. An estimate for small x.
Proof. Choose ρ 0 via Theorem 7.1 and let 0 < ρ < ρ 0 . Let {B i } k i=1 ⊆ R n be a regular simplicial conical partition. By Theorem 7.1 and the fact that ∆ 0 . . . , f k ) ∈ ∆ 0 k (γ n ). By (12), k i=1 R n f i T 0 f i dγ n = k(1/k 2 ) = 1/k. By the Fundamental Theorem of Calculus and (126), By using the invariance principle of [12, Theorem 1.10,Theorem 3.6,Theorem 7.1,Theorem 7.4] which transfers results from partitions of Euclidean space to low-influence discrete functions, Theorem 7.2 implies a weak form of the Plurality is Stablest Conjecture. While the following result is quite far from Conjecture 2 and might not be of immediate use to complexity theory, it is included to indicate a possible application of Theorem 7.2. Essentially, if we modify the exact application of the invariance principle that is used in [12, Theorem 7.1], then Conjecture 2 follows. However, by avoiding [12, Theorem 7.1], we must make very restrictive assumptions on the function f in Conjecture 2. Nevertheless, [12,Theorem 7.4] shows that the class of functions f described in Corollary 7.3 is nonempty.
Note that the most straightforward application of Theorem 7.2 only gives vacuous cases of Conjecture 2, in which 0 < ρ < ρ 0 (n, k). In particular, since Theorem 7.2 requires 0 < ρ < ρ 0 (n, k), by (12) we must take ε < 3kρ to get a nontrivial statement in Conjecture 2. In this case, the invariance principle [12,Theorem 3.6] gives τ with log τ = −C(log(ε)) 2 (1/ε), so that τ becomes a function of ρ. Since we provide a ρ with inverse exponential dependence on n, then τ also has inverse exponential dependence on n. Thus, no function f can satisfy the assumptions of Conjecture 2 in this case. To avoid this issue, we modify Conjecture 2 as follows.
Then part (a) of Conjecture 2 holds. From [12][ Theorem 7.4], this class of f is nontrivial.
Unfortunately, the proof of Theorem 7.2 fails for small negative ρ, as we now show.
Courant Institute, New York University, New York NY 10012 E-mail address: heilman@cims.nyu.edu