Applications of the `Ham Sandwich Theorem' to eigenvalues of the Laplacian

We apply Gromov's ham sandwich method to get (1) domain monotonicity (up to a multiplicative constant factor); (2) reverse domain monotonicity (up to a multiplicative constant factor); and (3) universal inequalities for Neumann eigenvalues of the Laplacian on bounded convex domains in a Euclidean space.

(2) (Restricted reverse domain monotonicity for Neumann eigenvalues) If in addition Ω ′ \ Ω has measure zero then λ N k (Ω) ≤ λ N k (Ω ′ ) for any k. These two properties are a direct consequence of Courant's minimax principle (see [Cha84]). The following two examples suggest that the domain monotonicity does not hold for Neumann eigenvalues in general.
Example 1.1. Let Ω ′ be the n-dimensional unit cube [ 0, 1 ] n . Then λ N 1 (Ω ′ ) = 1. However if Ω is a convex domain in [ 0, 1 ] n that approximates the segment connecting the origin and the point (1, 1, · · · , 1) then λ N 1 (Ω) ∼ 1/n. Example 1.2. Let p ∈ [ 1, 2 ] and B n p be the n-dimensional ℓ p -ball centered at the origin. Suppose that r n,p is the positive number such that vol (r n,p B n p ) = 1 and set Ω ′ := r n,p B n p . Then r n,p ∼ n 1/p and λ N 1 (Ω ′ ) ≥ c for some absolute constant c > 0 ([Sod08, Section 4 (2)]). If the segment in Ω ′ connecting the origin and (r n,p , 0, 0, · · · , 0) is approximated by a convex domain Ω in Ω ′ then λ N 1 (Ω) ∼ r −2 n,p ∼ n −2/p . In this paper we study the above two properties for Neumann eigenvalues of the Laplacian on convex domains in a Euclidean space. For two real numbers α, β we denote α β if α ≤ cβ for some absolute constant c > 0.
One of our main theorems is the following: Theorem 1.3. For any natural number k ≥ 2 and any two bounded convex domains Ω, Ω ′ in R n with piecewise smooth boundaries such that Ω ⊆ Ω ′ we have . As a corollary we get the following inner radius estimate: Let Ω ⊆ R n be a bounded convex domain with piecewise smooth boundary. For any k ≥ 2 we have where B 1 is a unit ball in R n .
We also obtain the opposite inequality to Theorem 1.3: Theorem 1.5. Let Ω, Ω ′ be two bounded convex domains in R n having piecewise smooth boundaries. Assume that Ω is symmetric with respect to the origin (i.e., Ω = −Ω) and Then for any natural number k ≥ 3 we have For general (not necessarily symmetric) Ω we have As a corollary of Theorems 1.3 and 1.5 we obtain for all k ≥ 2, which corresponds to the above properties (1) and (2) up to multiplicative constant factors. In [Mil09] E. Milman obtained the corresponding inequality for k = 1 (see (5.1)). Despite the fact that his inequality is independent of dimension, our two inequalities above involve dimensional terms. However log k bounds in the two inequalities are nontrivial (Compare with (5.2)). The case where p = 1 in Example 1.2 shows that the n 2 order in Theorem 1.3 cannot be improved. Probably there would be a chance to express the multiplicative constant factor in Theorem 1.3 in terms of the volume ratio v = vol Ω/ vol Ω ′ to avoid the dependence of dimension (see Question 5.3). As a special case where Ω = Ω ′ in Theorem 1.3 we obtain the following universal inequalities among Neumann eigenvalues : By 'universal' we mean it does not depend on the underlying domain Ω itself. Payne, Pólya, and Weinberger studied universal inequalities among Dirichlet eigenvalues ( [PPW55,PPW56]). Since then many universal inequalities for Dirichlet eigenvalues were studied (see [AB07]). For Neumann eigenvalues, Liu ([Liu14]) showed the sharp inequalities for any bounded convex domain Ω, which improves author's exponential bounds in k in [Fun13]. On the other hand, one can get for any bounded convex domain Ω ⊆ R n . This inequality follows from the combination of E. Milman's result [Mil09, Remark 2.11] and Cheng-Li's result [CL81] (see [SY94,Chapter III §5]). In fact E. Milman described the Sobolev inequality in terms of λ N 1 (Ω) and Cheng-Li showed lower bounds of λ N k (Ω) in terms of the Sobolev constant. The Weyl asymptotic formula says that the inequality (1.5) is sharp. In particular combining (1.4) with (1.5) we can obtain λ N k (Ω) k 2−2/n λ N k−1 (Ω). Comparing with this inequality our inequality (1.3) includes the dimensional term. However the dependence on k is best ever to author's knowledge. It should be mentioned that author's conjecture in [Fun13,Fun16] for any bounded convex domain Ω with piecewise smooth boundary.
In the proof of Theorems 1.3 and 1.5 we will use Gromov's method concerning a bisection of finite subsets by the zero set of a finite combination of eigenfunctions. It enables us to get lower bounds for eigenvalues of the Laplacian in terms of Cheeger constants and the maximal multiplicity of a covering of a domain (Proposition 3.1). We will try to find 'nice' convex partition in order to get 'nice' lower bounds for Cheeger constants of pieces of the partition.

Separation distance.
Let Ω be a bounded domain in a Euclidean space. For two subsets A, B ⊆ Ω we set dΩ (A, B) := inf{|x − y| | x ∈ A, y ∈ B}. We denote by µ the Lebesgue measure on Ω normalized as µ(Ω) = 1.
Theorem 2.2 ([Fun16, Theorem 1]). There exists an absolute constant c > 0 satisfying the following property. Let Ω be a bounded convex domain in a Euclidean space with piecewise smooth boundary and k, l be two natural numbers with l ≤ k. Then we have The case where k = l = 1 was first proved by Gromov and V. Milman without convexity assumption of domains ( [GM83]). Chung, Grigor'yan, and Yau then extended to the case where k = l ( [CGY96,CGY97]). To reduce the number l of subsets in Ω in a dimensionfree way we need the convexity of Ω (see [Fun16]).
2.2. Cheeger constant and eigenvalues of the Laplacian. For a Borel subset A ⊆ Ω and r > 0 we denote U r (A) the r-neighborhood of A in Ω. We define the Minkowski boundary measure of A as r .

Definition 2.3 (Cheeger constant). For a bounded domain Ω in a Euclidean space we define the Cheeger constant of Ω as
where the infimum runs over all non-empty disjoint two Borel subsets A 0 , A 1 of Ω.
Let µ be a finite Borel measure on a bounded domain Ω ⊆ R n and f : Let Ω be a bounded convex domain in a Euclidean space and assume that Ω satisfies the following concentration inequality for some r > 0 and κ ∈ ( 0, 1/2 ) : µ(Ω \ U r (A)) ≤ κ for any Borel subset A ⊆ Ω such that One can easily check that Theorem 2.5 has the following equivalent interpretation in terms of separation distance.
The latter statement can be found in [ 2.3. Voronoi partition. Let X be a metric space and {x i } i∈I be a subset of X. For each i ∈ I we define the Voronoi cell C i associated with the point x i as Note that if X is a bounded convex domain Ω in a Euclidean space then {C i } i∈I is a convex partition of Ω (the boundaries ∂C i may overlap each other). Observe also that if the balls {B(x i , r)} i∈I of radius r covers Ω then C i ⊆ B(x i , r), and thus diam C i ≤ 2r for any i ∈ I.

Gromov's ham sandwich method
In this section we explain Gromov's ham sandwich method to estimate eigenvalues of the Laplacian from below. Recall that the classical ham sandwich theorem in algebraic topology asserts that given three finite volume subsets in R 3 , there is a plane that bisects all these subsets ( [Mat03]). In stead of bisecting by a plane we consider bisecting by the zero set of a finite combination of eigenfunctions of the Laplacian in Gromov's ham sandwich method.
Let Ω be a bounded domain in a Euclidean space with piecewise smooth boundary and {A i } l i=1 be a finite covering of Ω; Ω = i A i . We denote by M({A i }) the maximal multiplicity of the covering {A i } and by h({A i }) the minimum of the Cheeger constants of A i , i = 1, 2, · · · , l.
Step 1. Use the Borsuk-Ulam theorem to get constants c 0 , c 1 , · · · , c l such that f : In fact, according to [ST42,Corollary], in order to bisect l subsets by a finite combination of f 0 ≡ 1, f 1 , · · · , f l , it suffices to check that f 0 , f 1 , · · · , f l are linearly independent modulo sets of measure zero (i.e., whenever a 0 f 0 + a 1 f 1 + · · · + a l f l = 0 over a Borel subset of positive measure, we have a 0 = a 1 = · · · = a l = 0). This is possible since the zero set of any finite combination of f 0 , f 1 , · · · , f l has finite codimension 1 Hausdorff measure ([BHH16, Subsection 1.1.1]).
Step 2. Put f + (x) := max{f (x), 0} and f − (x) := max{−f (x), 0}. Then we set g ± := f 2 ± . Note that 0 is the median of the restriction of g ± to each A i by Step 1. Apply Theorem 2.4 to get h g Step 3. Use Step 2 to get Recalling that g ± = f 2 ± and using the Cauchy-Schwarz inequality we have Since the zero set f −1 (0) has measure zero we get We therefore obtain l i=0 c 2 i ≤ (4M 2 /h 2 ) l i=1 c 2 i λ N i (Ω) and thus the conclusion of the proposition.
Remark 3.2. 1. In [Gro99] Gromov treated the case where Ω is a closed Riemannian manifold of Ricci curvature ≥ −(n − 1) and the covering consists of some balls B i of radius ε in Ω. In stead of considering the (1, 1)-Poincaré inequality in terms of Cheeger constants in Step 2 he proved that ε) is a constant depending only on dimension n and ε, and B i is the ball of radius 2ε with the same center of B i .
2. The above proposition is also valid for the case where Ω is a closed Riemannian manifold or a compact Riemannian manifold with boundary. In the latter case we impose the Neumann boundary condition.
As an application of Proposition 3.1 we can obtain estimates of eigenvalues of the Laplacian of closed hyperbolic manifolds due to Buser ([Bus80, Theorems 3.1, 3.12, 3.14]). In fact Buser gave a partition of a closed hyperbolic manifold and lower bound estimates of Cheeger constants of each piece of the partition.

Proof of main theorems
Let Ω, Ω ′ be two bounded convex domains in a Euclidean space. Throughout this section µ is the Lebesgue measure on Ω ′ normalized as µ(Ω ′ ) = 1.
Proof of Theorem 1.3. We apply Gromov's ham sandwich method (Proposition 3.1) to bound λ N k−1 (Ω) from below in terms of λ N k (Ω ′ ). To apply the proposition we want to find a finite partition {Ω i } l i=1 of Ω with l ≤ k − 1 such that the Cheeger constant of each Ω i can be comparable with λ N k (Ω ′ ). According to Theorem 2.2 we have for some absolute constant c > 0. We set R := (cn log k)/ λ N k (Ω ′ ). Suppose that Ω ′ includes k (4R)-separated points x 1 , x 2 , · · · , x k . By Theorem 2.7 we have diam Ω ′ ≤ c ′ nk/ λ N k (Ω ′ ) for some absolute constant c ′ > 0. Applying the Bishop-Gromov inequality we have for each i. If we rechoose c in (4.1) as a sufficiently large absolute constant so that (c log k)/c ′ ≥ 1 we get µ(B(x i , R)) ≥ 1/k n . Since B(x i , R)'s are 2R-separated this contradicts to (4.1).
Let y 1 , y 2 , · · · , y l be maximal 4R-separated points in Ω ′ , where l ≤ k − 1. Since is the Voronoi partition associated with {y i } then we have diam Ω ′ i ≤ 8R. Setting Ω i := Ω ′ i ∩ Ω we get Ω = l i=1 Ω i and diam Ω i ≤ 8R. Since each Ω i is convex, Proposition 2.6 gives h(Ω i ) ≥ 1/(8R). Applying Proposition 3.1 to the covering {Ω i } we obtain which yields the conclusion of the theorem. This completes the proof.
In order to prove Theorem 1.5 we prepare several lemmas.

Proof. Due to Theorem 2.2 we have
Sep Ω; 1 k n , .
In order to adapt to the hypothesis of Lemma 4.2 we use the following improvement of Borell's lemma.

Questions
In this section we raise several questions which concern this paper.