The fundamental gap of simplices

The fundamental gap conjecture was recently proven by Andrews and Clutterbuck: for any convex domain in $\R^n$ normalized to have unit diameter, the difference between the first two Dirichlet eigenvalues of the Laplacian is bounded below by that of the interval. In this work, we focus on the moduli spaces of simplices in all dimensions, and later specialize to the moduli space of Euclidean triangles. Our first theorem is a compactness result for the gap function on the moduli space of simplices in any dimension. Our second main result verifies a recent conjecture of Antunes-Freitas: for any Euclidean triangle normalized to have unit diameter, the fundamental gap is uniquely minimized by the equilateral triangle.


Introduction
Let Ω ⊂ R n be a convex domain. Let λ 1 and λ 2 be the first two eigenvalues of the Euclidean Laplacian on Ω with Dirichlet boundary condition. It is a classical result that 0 < λ 1 < λ 2 . The gap between λ 1 and the rest of the spectrum, is known as the fundamental gap of Ω. The gap function where d is the diameter of Ω. The gap function is a scale invariant: it is purely determined by the shape of the domain. Physically, if we consider heating the domain at some initial time and then keeping the boundary of the domain fixed at zero temperature, the fundamental gap determines the rate at which the overall heat in the domain vanishes as time tends to infinity. It is natural to ask the following question.
How does the shape of a convex domain affect the rate at which it loses heat over a long period of time?
The mathematical formulation of this question is: What is the relationship between the geometry of a convex domain Ω ⊂ R n and ξ(Ω)?
M. van den Berg [17] observed that for many convex domains, the gap function is bounded below by a constant. For example, consider a rectangular domain R ⊂ R 2 , Using separation of variables, it is straightforward to compute that the eigenfunctions and corresponding eigenvalues of the rectangle are φ j,k (x, y) = sin jxπ a sin kyπ b , λ j,k = j 2 π 2 a 2 + k 2 π 2 b 2 , j, k ∈ N.
Making the additional assumption b ≤ a, one computes the gap function of the rectangle R, If we then think about the gap function on all possible rectangles R, we see that the square uniquely maximizes the gap function with ξ(Square) = 6π 2 .
On the other hand, if a rectangle collapses to a segment, by letting b ↓ 0, then ξ ↓ 3π 2 . An even more elementary example is the segment. The gap function on any (finite) segment [a, b] with a < b is Perhaps based on this intuition, Yau formulated the fundamental gap conjecture, in [18] which was recently proven by Andrews and Clutterbuck [1].
Theorem 1 (Andrews-Clutterbuck). For any convex domain in R n , the gap function is bounded below by 3π 2 .
This result shows that among all convex domains, the gap function is minimized in dimension 1. If the gap function is restricted to a certain moduli space of convex domains, what are its properties?
In this work, we focus on the gap function restricted to the moduli space of nsimplices and in particular, the moduli space of Euclidean triangles. Recall that an n-simplex X is a set of n+1 vectors {v 0 , · · · , v n } in R n such that v 1 −v 0 , · · · , v n −v 0 are linearly independent. The convex domain    n j=0 t j v j n j=0 t j = 1, t j ≥ 0 for 0 ≤ j ≤ n    defined by X is bounded with piecewise smooth boundary. For the sake of simplicity, we don't distinguish the simplex X with the domain it defines. The moduli space of n-simplices is the set of all similarity classes of n-simplices; it is parametrized by the set of n-simplices with diameter equal to one. We note that in case n = 2, this theorem is straightforward to deduce from the main result of [6].
Theorem 2. Let Y be an n − 1 simplex for some n ≥ 2. Let {X j } j∈N be a sequence of n simplices each of which is a graph over Y . Assume the height of X j over Y vanishes as j → ∞. Then ξ(X j ) → ∞ as j → ∞. More precisely, there is a constant C > 0 depending only on n and Y such that ξ(X j ) ≥ Ch(X j ) −4/3 , where h(X j ) is the height of X j .
Since any triangle with unit diameter is a graph over the unit interval, this theorem implies that there exists at least one triangle which minimizes the gap function on the moduli space of triangles. The moduli space of triangles is the set of all similarity classes of triangles, which we identify with M ∼ = (x, y) ∈ R 2 : y > 0, where the vertices of a triangle in each similarity class are (0, 0), (1,0) and (x, y).
The following result shows that the gap function on triangular domains is more than twice as large as the gap function on a generic convex domain; the theorem was conjectured in [2].
Theorem 3. For any triangle T with unit diameter, where equality holds iff T is equilateral.
Let us recall the famous question posed by M. Kac [9]: Can one hear the shape of a drum? The resonant tones of a domain are in bijection with the eigenvalues of the Euclidean Laplacian with Dirichlet boundary condition. Therefore, with a perfect ear that is capable of registering all tones, one can hear the spectrum, that is, the set of all eigenvalues. Kac's question is then mathematically reformulated as follows.
If two domains in R 2 have the same spectrum, do the domains also have the same shape? A negative answer to Kac's question was demonstrated by Gordon, Webb and Wolpert [7], who showed that there exist isospectral planar domains which are not isomorphic. On the other hand, Durso [5] proved that if the two domains are triangles in R 2 , and they have the same spectrum, then they must be the same triangle. The proof uses the entire spectrum, so we can reformulate her result as follows.
With a perfect ear, one can hear the shape of a triangle [5]. In practice, however, one does not have a perfect ear. That is, one may only detect a finite portion of the spectrum. Our Theorem 3 implies: Amongst all triangles, one can hear the shape of the equilateral triangle. In particular, the equilateral triangle can be heard with a realistic ear, because Theorem 3 demonstrates that the gap function uniquely distinguishes the equilateral triangle within the moduli space of all triangles. In fact, we expect that it is possible to distinguish triangles based on a finite number of eigenvalues. This is supported by numerical data in [3], which shows that one expects that triangles are uniquely determined by their first three eigenvalues.
Our work is organized as follows. The compactness result for simplices is proven in §2; in §3 this is refined to prove that Theorem 2 holds for all sufficiently "thin" triangles. In §4, we prove that the equilateral triangle is a strict local minimum for the gap function on the moduli space of triangles, and in §5 we determine a lower bound for the radius of the neighborhood in the moduli space of triangles on which the equilateral triangle is a strict local minimum. Finally, in §6, we provide an algorithm to complete the proof of Theorem 2. Concluding remarks and conjectures are offered in §7.
for k ≥ 1 and l ≥ 0, where ϕ j achieves the infimum for k = j (and as above φ 0 ≡ 0 and ϕ −1 ≡ 0). Finally, throughout this paper we will use the following notations: for a function f (t) and fixed k ≥ 0, To prove Theorem 2, we show that if a sequence of n-simplices collapse, there exists C > 0 such that ξ(X j ) ≥ Ch(X j ) −4/3 , where the height h of the simplex (defined in the arguments below) vanishes as j → ∞. For simplicity in notation, let us drop the subscript. We may assume that the simplex is defined by the points are the standard basis of R n . In other words, p 0 , . . . , p n−1 are contained in the span of {e i } n−1 i=1 . The collapse is described by |p n n | → 0. In fact, we may assume without loss of generality that the simplex is contained in the set of points x k e k , x n ≥ 0 .
Then, for any point x k e k , the height of x h(x) := x n . The height of the simplex itself is defined to be h = h(X) := h(p n ) = p n n . Since the simplex collapses, we assume in the remaining arguments that h < 0.1.
Let λ i , i = 1, 2, be the first and second Dirichlet eigenvalues of X with corresponding eigenfunctions φ i such that X φ 2 i = 1. In the following claim, we demonstrate the quantitative estimate that at least 90 % of the mass of the eigenfunctions φ 1 and φ 2 is contained in a cylinder around p n intersected with X. We call this estimate "cutting corners" because it shows that we may "cut off the corners" and use the cylinder to estimate the gap. Let and let B n−1 ch 2/3 (p) be the (n−1) dimensional ball in the space spanned by e 1 , · · · , e n−1 . The constant c will be chosen later. We define U to be the intersection of the cylinder with base B n−1 ch 2/3 (p) and height h with X, U := B n−1 where I h is the interval of length h. Let and let Claim: There exists a constant A which depends only on n and Y such that if c > A and h ≤ 1 2c Proof of Claim: We shall begin by assuming By definition of the simplex as the convex hull of its defining points, since p 0 , . . . , p n−1 are contained in the span of e 1 , . . . , e n−1 , the diameter of the simplex is 1, and h ≤ 0.1, it follows that By the one dimensional Poincaré inequality and since U φ 2 On the other hand, X contains a cylinder where h 2/3 Y is the base scaled by h 2/3 . One computes explicitly where C 2 is the second Dirichlet eigenvalue of Y . Consequently, (2.3) and (2.4) imply that for i = 1, 2, where the final inequality follows since h < 1 10 . On the other hand, , then c > A and h ≤ 1 2c Consider the so-called "drift Laplacian" ∆ U on U with respect to the weight function f = −2 log φ 1 , ∆ U := ∆ + 2∇ log φ 1 ∇. Let µ be the first non-zero Neumann eigenvalue of ∆ U on U , and let Then, ψ satisfies ∆ψ + 2∇ log φ 1 ∇ψ = −(λ 2 − λ 1 )ψ. Let Thus, by the weighted variational principle, sinceψ satisfies (2.6), We have Using the claim we have, .

Theorem 2 is true for short triangles
Refining estimates from the proof of Theorem 1, we demonstrate that if a triangle is sufficiently "short," its fundamental gap is strictly larger than 64π 2 /9.
Proof: Define where the constant c will be specified later. The main idea, as in the proof of Theorem 2, is that λ 2 − λ 1 is well approximated by the first positive Neumann eigenvalue of U . Assume the eigenfunctions φ i for λ i satisfy and let Noting that the weighted variational principle for the first positive Neumann eigenvalue µ(U ) of the drift Laplacian ∆ U with weight function −2 log φ 1 gives We compute the denominator By Corollary 1 of [14] and Corollary 1.4 of [1], which implies Since φ 1 and φ 2 are L 2 orthogonal, which by the Cauchy Schwarz inequality and definition of β gives Consequently, .
Proceeding by contradiction, we assume By trigonometry, T contains a rectangle By domain monotonicity, The height of V is at most ]. By the one dimensional Poincaré inequality for i = 1, 2, Since Since h < 0.1, we have which simplifies to Since we assume h ≤ 0.005, then for any c < 34, ch 2/3 < 1. In particular, fixing c = 10, we compute that (3.3) gives for any h ≤ 0.005, Then, Since we compute that for any h ≤ 0.005,

The equilateral triangle is a strict local gap minimizer
The main result of this section demonstrates that the equilateral triangle is a strict local minimum for the gap function on the moduli space of triangles. In the proof, we consider all possible linear deformations of the equilateral triangle and demonstrate that in any direction, such a deformation strictly increases the gap function. Figure 1. Linear deformation of a triangle.

4.1.
Linear deformation theory. Let T be a triangle with vertices (0, 0), (1, 0), and z = (j, k), and side lengths A ≤ B ≤ 1. Consider a deformation to the triangle T (t) which has vertices (0, 0), (1, 0), and z + tx, where The direction of the deformation is given by (a, b), while the magnitude is given by t ≥ 0. The linear transformation which maps the triangle T to the triangle T (t) is represented by the matrix We may view the linear transformation T → T (t) as a change of the (Euclidean) Riemannian metric on R 2 . In other words, T (t) is isomorphic to T with the metric, We compute Thus.
If the eigenvalues of the original triangle and the deformation triangle are respectively λ i and λ i (t), then they satisfy where γ ± are the eigenvalues of g −1 . It follows that We compute Substituting the values of A, B, and D gives in general The relationship between integration over T (t) and T differs by a linear factor, where throughout this paper, integration is with respect to the standard Lebesgue measure dxdy on R 2 . The Laplace-Beltrami operator associated to a Riemannian metric (in dimension n) is so one computes the Laplacian for the deformation metric g is Henceforth we shall use ∆ 0 = ∂ 2 x + ∂ 2 y for the Euclidean Laplacian, ∆ for the Laplacian associated to a deformation metric, and ∇f = (f x , f y ) the gradient (with respect to the standard Euclidean metric). In §6, we shall use this linear deformation theory to complete the proof of Theorem 2. Presently, we specialize to the equilateral triangle which we call T , and whose vertices are (0, 0), (1, 0), and 1 2 , A triangle obtained by a linear deformation of T , with vertices (0, 0), (1, 0), and 1 2 , is equivalent to T with the metric We compute, The associated Laplace operator and (4.5) By the variational principle since λ 1 is smooth, the first eigenvalue for the T (t), which we write as λ 1 (t) satisfies where φ 1 is an eigenfunction for λ 1 with unit L 2 norm. If λ 2 is simple, then we also have where φ 2 is an eigenfunction for λ 2 with unit L 2 norm. In general, λ 2 is not differentiable because the second eigenspace may have dimension 2; this is the case for the equilateral triangle. Nonetheless, we may use the variational principle to show that the equilateral triangle is a strict local minimum for the gap function restricted to the moduli space of triangles.
Theorem 4. The equilateral triangle is a strict local minimum for the gap function on the moduli space of triangles.
We will prove the theorem by applying the following proposition together with explicit calculations for the eigenvalues and eigenfunctions of the equilateral triangle.
Proposition 2. For any deformation of the equilateral triangle which preserves diameter, for the corresponding L 1 , Proof that Proposition 2 implies Theorem 4: Let φ 1 and φ 2 be eigenfunctions for the first two Dirichlet eigenvalues λ 1 and λ 2 , respectively, for the equilateral triangle T . Assume the eigenfunctions have unit L 2 norm on T . Let f 1 and f 2 be eigenfunctions for the first two Dirichlet eigenvalues of T (t). Let where above and indeed throughout this paper, integration is over the equilateral triangle T unless otherwise indicated. For simplicity, in this section we shall use L 1 to denote L 1 | t=0 . Since A is a linear transformation from T to T (t), f 1 and f 2 are orthogonal with respect to dxdy, so by the convergence of f 1 → φ 1 , Note that so by the variational principle, Since these functions are uniformly bounded in C k for any fixed k, the Laplace operator on T (t) Then, Consequently, Then, (4.8) and (4.9) imply (4.10) Since the deformation preserves diameter, we may re-write (4.10) as We can always construct a sequence of eigenfunctions f 2 which converge in C 2 to some eigenfunction φ 2 for λ 2 with φ 2 2 = 1. Consequently, Since for all φ 2 , we have ξ(T ) < ξ(T (t)) for all t sufficiently small. Finally, we note that we need only consider deformations in directions which preserve the diameter because the gap function is scale invariant. We have therefore reduced the theorem to verifying explicit calculations involving the eigenfunctions and eigenvalues of the equilateral triangle.

4.2.
Eigenfunctions and eigenvalues of the equilateral triangle. In 1852, Lamé computed the eigenfunctions and eigenvalues of the equilateral triangle by (real) analytically extending the eigenfunctions to the plane using the symmetry of the equilateral triangle [12], [10], [11]. The eigenvalues are given by the general formula (4.14) Since the first Dirichlet eigenvalue is always simple, it follows from Corollary 2 of [15] that the first L 2 normalized eigenfunction of the equilateral triangle T is Proposition 3.
Proof. The standard angle addition and subtraction identities for sine and cosine show that, The last inequality follows from the identity We compute

4.2.2.
The second eigenspace of the equilateral triangle. The second Dirichlet eigenvalue of the equilateral triangle is given by (4.12) with m = 1 and n = 5 (or with m = −1 and n = 4), This eigenspace has dimension two. An L 2 orthonormal basis of eigenfunctions is given by and v(x, y) = 2 The following calculations will play a key role in the proof of Theorem 2. .

4.2.3.
The third and higher eigenspaces of the equilateral triangle. The third (distinct) eigenvalue is given by (4.12) with m = n = 6, The eigenspace has dimension one. The eigenfunction is
As previously observed, we need only consider those deformations in directions which preserve diameter, and by symmetry, we need only consider those directions θ with cos θ ≥ 0. These are deformations in directions θ ∈ [−π/2, −π/6], so the direction vector (a, b) satisfies a 2 + b 2 = 1 , a ≥ 0, and a + √ 3b ≤ 0. We compute the minimum of So, we determine the minimum of subject to the constraints Introducing the polar coordinates, cos(t) := α, sin(t) := β, we compute that I is minimized for and the minimum is Remark 1. In the proof of the theorem, we have shown that for any triangle T ∈ M with vertices (0, 0), (1, 0), and (x, y) In the arguments below, we use the calculations in §2 to precisely estimate the error O(t 2 ).

Theorem 2 is true for almost equilateral triangles
The following proposition is the last step we need to reduce the proof of Theorem 2 to finitely many numerical calculations.
Since λ 1 is differentiable, we have By the variational principle, , which simplifies to since integration over T and T (t) differ by linear factors which cancel in the numerator and denominator, and φ 2 1 = 1. We then have where L = L 1 + L 2 is defined in (4.4, 4.5). So, we compute directly Thus, we have made explicit We have We estimate using the calculations for the first eigenfunction of the equilateral triangle and a 2 + b 2 = 1, which for t ≤ 0.0004 gives

5.1.
Estimates for the second eigenspace. Since λ 2 of the equilateral triangle is not differentiable, the estimates for its error term require a bit more work. The main idea is to expand the first two eigenfunctions for the linearly-deformed triangle using the orthonormal basis of eigenfunctions for the equilateral triangle. We then use the Poincaré inequality and our explicit calculations for the eigenfunctions of the equilateral triangle to estimate the error.
5.1.1. The first eigenfunction of the linearly deformed triangle. Our eventual goal is to estimate λ 2 (t) from below. To accomplish this, we require not only estimates for the second eigenspace of the linearly deformed triangle, T (t), but also estimates for its first eigenfunction. Let f be the first eigenfunction of T (t) and write As usual, integration is over the equilateral triangle T with respect to the standard measure dxdy, and we use || · || to denote the L 2 norm over T . Since we assume t ≤ 0.0004, by (4.2) We compute Since φ 1 g = 0, by the variational principle (2.1) which gives the Poincaré inequality for g, To estimate ||∇g||, we use the definition of f and g to compute By definition of g and (5.2), We compute using integration by parts and then substituting (5.6, 5.7) g(∆ 0 + λ 1 )g = − |∇g| 2 + λ 1 g 2 = −tα 1 g 2 − gLφ 1 − t gLg, which gives By definition of L, and since a 2 + b 2 = 1, for any function ψ which vanishes on ∂T , we have and since we always have ||ψ x || ≤ ||∇ψ||, ||ψ y || ≤ ||∇ψ||, and −1 ≤ b ≤ 1, we have for any ψ which vanishes on ∂T.
Moreover, we have the estimate for ||g||, The second eigenspace. Let F be an eigenfunction in the second eigenspace of T (t), and assume ||F || = 1, where as usual, the L 2 norm is taken over the equilateral triangle T . Expanding F in terms of the eigenfunctions of the equilateral triangle, where ϕ is an eigenfunction for λ 2 , and G satisfies Then, we have

so by the Cauchy inequality
By definition of F and G, Combining these, we have so we obtain the estimate for A, Since G is orthogonal to the first two eigenspaces, the variational principle for λ 3 gives λ 3 ≤ |∇G| 2 G 2 , which implies the Poincaré inequality for G, We estimate G in the same spirit as g. We compute since So, To estimate ||∇G|| and hence ||G|| by the Poincaré inequality (5.15), we use the above calculation together with integration by parts (as we did with g), By definition of F and (5.13), this is On the other hand, integrating by parts gives Combining this with the above calculation, we have The estimate for L (5.9) and the Cauchy inequality imply By the Poincaré inequality for G (5.15), This gives the estimate Expanding and simplifying we have Recalling the estimate (5.14) for A, Collecting the ||G|| terms, This gives the estimate from above for ||G|| (5.16) At this point, we may substitute estimates for every term except ||Lϕ||, which we now estimate. By definition of L, By the triangle inequality for L 2 and since t ≤ 0.001, Since ϕ is an L 2 orthonormal eigenfunction for λ 2 , ϕ = αu + βv, α 2 + β 2 = 1. Based on these estimates, we shall use the variational principle to estimate λ 2 (t) from below. Since F − Aφ 1 = ϕ + tG is orthogonal to φ 1 , the variational principle for λ 2 gives We compute the numerator to be The last two terms are By definition of φ 1 and integration by parts, these are The first integral vanishes since ϕ and G are orthogonal to φ 1 . So, the numerator of (5.20) is Thus, the variational principle for λ 2 implies which gives the estimate for λ 2 (t), This implies Substituting (5.13), we have By definition of F = ϕ + Aφ 1 + tG and integration by parts, This gives Incorporating estimate (5.1) for λ 1 (t), we have Our calculations from the proof of Theorem 2 and the estimate (5.1) of |O 1 (t 2 )| imply Recall the calculation We estimate using the Cauchy inequality, ||ϕ x || and ||ϕ y || ≤ ||∇ϕ|| with ||∇ϕ|| 2 = λ 2 , Since we assume t ≤ 0.0004, we have We estimate the remaining terms using the Cauchy inequality, our estimates for ||Lϕ||, ||Lφ 1 ||, G, ||∇G||, and the general estimate (5.9) for L, This is satisfied for any t ≤ 0.0004.

Proof of Theorem 2
By our preceding results and continuity of the eigenvalues, we may now complete the proof of Theorem 3 by computing the first two eigenvalues of a large but finite number of triangles. 6.1. Continuity estimate. The following calculation is based on the linear deformation theory at the beginning of Section 4. We use T (x, y) to denote a triangle with vertices (0, 0), (1, 0), and (x, y), and we use λ i (x, y) to denote its i th Dirichlet eigenvalue, and ξ(x, y) to denote its fundamental gap, If a triangle T (x * , y * ) satisfies then by (4.1), Therefore, for each triangle T (x, y) at which we compute numerically we may use (6.1) to determine a neighborhood of triangles satisfying without numerically computing the eigenvalues of the triangles in this neighborhood. Consequently, we have reduced the problem to numerically computing the fundamental gap of finitely many triangles and using the following algorithm.
6.2. Algorithm. The main idea of the algorithm is to use the preceding calculations to compute, to sufficient numerical accuracy, the first two eigenvalues of a finite grid of triangles and use this grid together with the continuity estimate to demonstrate that the gap of all triangles lying outside the cases covered by Propositions 1 and 4 is strictly larger than that of the equilateral triangle. In particular, it follows from Propositions 1 and 4, the invariance of the gap function under scaling and symmetry that we need only compute for those triangles with vertices (0, 0), (1, 0) and (x, y), such that the following inequalities hold. i.1 x 2 + y 2 ≤ 1, by invariance of the gap function under scaling. i.2 0.5 ≤ x ≤ 1, by symmetry. i.3 0.005 ≤ y ≤ 1, by Proposition 1.
6.2.1. The steps of the algorithm. We begin with the triangle whose vertices are (0, 0), (1, 0) and (0.5, 0.005); this is step 0. Next, in steps 1-2, we compute using (6.1) the radius t of the neighborhood around which the gap is strictly larger than 64π 2 /9. We then increase the x-coordinate in step 3, and check that the inequalities i.1-i.4 hold. If so, we repeat the calculations in steps 1-2 for the triangle whose third vertex is at the same height y but has been translated in the positive xdirection (to the right). We repeat steps 1-3 until the x coordinate is large enough so that one of the inequalities i.1-i.4 fails; then we proceed to step 4. In step 4, we return the x-coordinate to 0.5 and increase the y-coordinate and check that the inequalities i.1-i.4 hold. We then continue repeating steps 1-4.
6.3. The numerical methods. The numerical computation of the eigenvalues were done by Timo Betcke using the Finite Element Method FreeFEM++ [8]. For efficiency, the calculations are made at each step but not stored, with the exception of t 0,j which must be stored until it is replaced by t 0,j+1 . To demonstrate the behavior of the gap function numerically, Timo plotted the logarithm of the gap function in the figure below. The grid points are parametrized so that each grid point corresponds to a triangle with vertices (0, 0), (1,0) and (x, y) where Hence, the equilateral triangle corresponds to ν = τ = 1. Recently, Laugesen and Siudeja [13] proved an interesting related result. For n = 1, (6.2) is well known. The case n = 2 can be deduced from Theorem 3 as follows. By Theorem 3 and (6.2) with n = 1, The existence and identity of a gap-minimizing simplex is a challenging open problem. Based on our results, we expect the following.
Conjecture 2. Let M n be the moduli space of all n-simplices with unit diameter. For n ≥ 2, the regular simplex defined by points p 0 , p 1 , . . . , p n ∈ R n such that |p i − p j | = 1 for 0 ≤ i = j ≤ n uniquely minimizes the gap function on M n .
There are several difficulties to be addressed. A subtle problem is the behavior of the gap of a family of collapsing simplices when several directions collapse simultaneously. Is it possible that competing collapsing directions may result in a gap which stays bounded or converges to that of the interval as simplices collapse? Numerical calculations would provide insight into what one might expect; combining classical techniques with modern computation may produce interesting new results.
We end this paper with a brief discussion of the similarities and differences between the behavior of the gap function on convex domains and the gap function restricted to the moduli space of n-simplices. In the fundamental work of [16] and subsequent papers [19], [20] culminating in the proof of the fundamental gap conjecture [1], the general method is to compare the eigenvalue estimate in higher dimensions to the eigenvalue estimate on a one dimensional manifold. The minimum gap for all convex domains can be asymptotically approached by thin tubular domains, and the minimum is achieved in dimension one. We pose the natural question: Is this minimum unique? More precisely, we make the following conjecture.
In the case of triangular domains, the gap function is uniquely minimized by the equilateral triangle. It would be interesting to extend the beautiful works in the spirit of [16] and [1] to compare the eigenvalue estimate in higher dimensions to the eigenvalue estimate in dimensions greater than one. In particular, it would be interesting to compare the eigenvalue estimate to that on the equilateral triangle or other computable planar domains.