INVARIANT WEDGES FOR A TWO-POINT REFLECTING BROWNIAN MOTION AND THE \HOT SPOTS" PROBLEM

We consider domains D of R d , d (cid:21) 2 with the property that there is a wedge V (cid:26) R d which is left invariant under all tangential projections at smooth portions of @D . It is shown that the di(cid:11)erence between two solutions of the Skorokhod equation in D with normal reﬂection, driven by the same Brownian motion, remains in V if it is initially in V . The heat equation on D with Neumann boundary conditions is considered next. It is shown that the cone of elements u of L 2 ( D ) satisfying u ( x ) − u ( y ) (cid:21) 0 whenever x − y 2 V is left invariant by the corresponding heat semigroup. Positivity considerations identify an eigenfunction corresponding to the second Neumann eigenvalue as an element of this cone. For d = 2 and under further assumptions, especially convexity of the domain, this eigenvalue is simple.


Introduction
The "hot spots" property of a bounded connected open domain D ⊂ R d refers to the location of the extrema of eigenfunctions corresponding to the second eigenvalue of the Laplacian on D with Neumann boundary conditions. Among the various statements associated with this property [1,3,5,10,11] let us mention: (HS) Every eigenfunction corresponding to the second Neumann eigenvalue attains its extrema solely on the boundary.
(HS') There exists an eigenfunction corresponding to the second Neumann eigenvalue which attains its extrema solely on the boundary.
The name comes from considering the heat equation on D: Under weak assumptions on D an eigenfunction expansion for the solutions of the heat equation is available. Since the eigenfunction corresponding to the first eigenvalue is a constant, the spatial locations of extrema of the solution for a typical initial condition are governed at large values of t by the second eigenspace. The assertion is therefore on the "hottest" and "coldest" spots in D.
Although it has been shown that (HS) holds for special domains by direct calculation, there are large classes of domains about which little is known. For example, for a planar simply connected domain with smooth boundary and no line of symmetry, there is no known criterion (to the author's best knowledge) to check whether (HS) is satisfied, other than calculation when it is possible. In [5] an example was first found of a domain that does not satisfy (HS), and in [3] an example was found of a domain for which the second Neumann eigenvalue (which we denote by µ 2 in what follows), is simple, and the corresponding eigenfunction attains both its strict extrema in the interior. See [1] for a conjecture that (HS) holds for all convex planar domains, and [5] for a conjecture that it holds for planar domains with at most one hole.
Bañuelos and Burdzy [1] were the first to identify rich classes of domains for which (HS) or (HS') hold. They use probabilistic techniques, and in particular, in one of the methods they develop, they consider a coupling of two normally reflecting Brownian motions driven by the same unconstrained Brownian motion. They consider planar domains and show that if all lines tangential to the boundary at its smooth portions form angles with a fixed axis within a range of less then π/2, then the difference between the processes will form an angle with this axis within this range for all times, if it does initially. They show that as a consequence, solutions to the Neumann heat equation are monotone along all lines within this range of angles when initialized appropriately, and therefore a similar statement holds for at least one eigenfunction corresponding to µ 2 . Hence the extrema of this eigenfunction are obtained on the boundary and (HS') holds. Under further assumptions they show that µ 2 is simple hence (HS) must hold. Jerison and Nadirashvili [10] have established the hot spots property for planar domains with two axes of symmetry.
In dimension greater than two, the only domains known to satisfy the hot spots property, other than domains with special symmetry, are those of the form D × [0, a] (see Kawohl [11]), or more generally D × D (see [1]). One of our two goals is provide new classes of domains in higher dimension satisfying (HS'). What might naively be expected to be the extension of the result of [1] regarding the coupled Brownian motions, does not hold. For example, the condition that all unit normals to ∂D at its smooth portions form scalar product with a fixed vector within the range of (− , ), does not guarantee that there is an invariant set for the coupled processes, in the sense described above, no matter how small > 0 is. We show that the approach of [1] can be generalized in a different way. Our assumption on D is that it is piecewise smooth with "convex corners" (see Condition 2.1). We then assume there is a wedge V ⊂ R d (see Definition 2.1) that is left invariant under all projections onto subspaces tangential to ∂D at smooth portions. We prove that two reflecting Brownian motions coupled as described above have difference in V if initialized there. As a result, the Neumann heat semigroup leaves the following cone of L 2 (D) invariant: Invoking positivity considerations the following is shown (see Theorem 3.1).
Assume that for some γ ∈ R d one has v, γ > 0 for all v ∈ V . Then there is an eigenfunction corresponding to the second Neumann eigenvalue attaining both strict extrema on the boundary.
(HS) follows from (HS') whenever it is known that µ 2 is simple. In a sense, it is typical that µ 2 is simple, as can be seen e.g. in [15]. It is known also that for simply connected planar domains, the multiplicity is at most two [16]. However, for a given domain it is in general hard to determine whether an eigenvalue is simple. An exception is a result that appears in [1], where it is shown that µ 2 is simple for convex planar domains for which the diameter to width ratio exceeds a certain number.
In this paper we identify a new class of planar domains for which µ 2 is simple. It is a subclass of the planar domains for which we prove that (HS') holds (with V = R 2 + ). As a result, they satisfy (HS). We show (see Theorem 4.1) Let D be a planar open domain bounded between the graphs of two C 2 increasing functions, one of which is convex and the other concave, such that D is bounded. Assume that for each corner point Ξ of the domain, B r (Ξ) ∩ D is a sector for some r > 0, of angle within [π/4, π/2). Then the second Neumann eigenvalue on D is simple.
It is conjectured in [1] (p. 5) that µ 2 is simple for all convex planar domains with diameter to width ratio greater than √ 2. The above statement addresses a subclass of this class of domains.
The rest of this paper is organized as follows. In Section 2 we make the assumption of existence of an invariant wedge V , and show that for a pair of solutions to the Skorokhod problem for an arbitrary continuous function, the difference is kept in V . Using results of [14] this is shown to imply a similar statement to semi-martingale reflecting Brownian motions. In Section 3 we exploit general positivity considerations to show that there must exist a second eigenfunction in the cone (2). Section 4 establishes simplicity of µ 2 for a class of planar convex domains. In the appendix we provide examples of three dimensional domains satisfying (HS').

An invariance property
In this section we use the results of Lions and Sznitman [14] that guarantee the existence of a unique solution to the Skorokhod problem for arbitrary continuous paths. We prove a certain invariance property in continuous paths space, which then translates to a property of semimartingale reflecting Brownian motions.
We will always consider domains that satisfy the following. For domains satisfying Condition 2.1 we will consider the vector field n of unit inward normals. The (not necessarily single-valued) vector field n is defined on the boundary ∂D of D as follows. For x ∈ ∂D i let n x,i denote the unit inward normal to ∂D i at x. Then for x ∈ ∂D we let We remark that the definition of [14] (equation (1) p. 514) of a vector field for a more general class of domains reduces to the above definition for the domains considered here.
For n ∈ N , the n-projection π n : R d → R d is defined as The most significant assumption we make is the following.

Condition 2.2 There exists a wedge
We will say that a wedge V is invariant for D if it satisfies Condition 2.2. We next formulate an equivalent to Condition 2.2.

Condition 2.2 There exists a wedge
where m is any inward normal to ∂V at v. Let v ∈ ∂V and let m be an inward normal to ∂V at v.
By assumption, π n v ∈ V for any n ∈ N , hence by convexity π n v − v, m ≥ 0. It follows that m, n n, v ≤ 0, hence Condition 2.2 holds (with the same set V ).
Next, let V satisfy Condition 2.2 . We will show that for v ∈ V and n ∈ N one has π n v ∈ V . If we assume the contrary then for some v ∈ V and n ∈ N we have π n v ∈ V c . Since V is convex with non-empty interior, every neighborhood of v contains a point in the interior. Since V c is open and π n continuous, there is a point w ∈ V o for which u . = π n w ∈ V c . The convexity of V implies that the line segment wu intersects ∂V at exactly one point, say z. There must exist an inward normalm to ∂V at z such that m, w − z > 0, and consequently m, u − z < 0. Note however that π n w = π n z = u. Hence m, π n z − z < 0 and it follows that m, n n, z > 0, in contradiction with Condition 2.2 . Therefore Condition 2.2 holds.
The definition of a solution to the Skorokhod problem for an arbitrary continuous function follows [14]. The notation |h| t is for the total variation of a function h : The following result is a special case of [14] Theorem 2.2 (see also [14] Remark 2.4 regarding domains with convex corners; the uniform exterior sphere condition obviously holds). Let w ∈ C([0, ∞), R d ) be such that w(0) = 0 and let x 0 , y 0 ∈D be given. By Theorem 2.1 there exists a unique solution to the Skorokhod problem (x 0 + w, D, n) [resp., (y 0 + w, D, n)] which we denote by (x, l x ) [resp., (y, l y )]. Thus for all t ≥ 0 and note that it is of bounded variation on any bounded interval.
Note that every wedge may occupy no more than a half space i.e., there must exist a γ ∈ R d for The following condition is slightly stronger.
Let V be some set satisfying Condition 2.2. We define on V c a vector field m as follows. For Note that the function is well defined since the normal is unique.

Lemma 2.1 Let u : [s, t] → V c be continuous and of bounded variation. Then
Proof: We first show that the integral on the right is well defined by showing that m(u(θ)) is in fact continuous in θ. It is enough to show that m(u) is continuous in u. The proof of this fact is elementary and for completeness we have included it in the appendix (see Lemma 5.1).
. It is elementary to show that for any r ∈ R d , |r| = 1, r . This shows that ∇ψ(x) is well defined and is equal to −m(x). Since m is continuous ψ is C 1 , and it follows that which proves the lemma.
Below, we borrow some ideas from the proof of [8] Theorem 2.2.

Proof of Theorem 2.2:
Let u ∈ ∂V and m = m(u). We first show that for all n ∈ N m, n n, u ≤ − m, n 2 .
Let v = u + m. Then it is easy to see that v ∈ ∂V and that m is an inward normal to ∂V at v. By Proposition 2.1, Condition 2.2 holds. Hence m, n n, v ≤ 0, n ∈ N and the estimate (7) follows.
Assume that the conclusion of the theorem does not hold. Note that δ s = 0 implies δ t = 0, t > s. Note also that by Since the argument is similar in both cases, we only consider case (a).
We close this section by showing an implication to reflecting Brownian motion. On a complete probability space (Ω, F, P ) with an increasing family of sub σ-fields The following result is proved in [14] (condition [14](9) holds by Remark [14] 3.9).

Theorem 2.3 (Lions and Sznitman) Let D satisfy Condition 2.1 and let x 0 ∈D be given. Then there exists a unique continuous
There exists an R d -valued continuous bounded variation process L t such that for all t ≥ 0 a.s. X t ∈D, The conclusion of Theorem 2.3 is equivalent to the statement that for a.e. ω ∈ Ω, (X, L) solves the Skorokhod problem (x 0 + W, D, n) (see e.g., Remark 3.2 of [14]).
Note that by definition X and Y are driven by the same process W , but have different initial conditions, x 0 and y 0 .
From Theorems 2.2 and 2.3 we obtain the following.
In what follows we will denote by P x,y the probability measure on (Ω, F, (F t )) for which P x,y [(X 0 , Y 0 ) = (x, y)] = 1. P x will denote the restriction of P x,y to the σ-field generated by X · . E x,y and respectively, E x will denote expectation with respect to P x,y and P x .

Remark:
In [8] Lipschitz continuity of the Skorokhod map in path space is shown to follow from the existence of a certain convex set. Condition 2.2 is in a sense analogous to Assumption 2.1 of [8] and the condition in Lemma 2.1 there. Also, the equivalence between Conditions 2.2 and 2.2 is a reminiscent of the results of Section 2.5 of [9].
What provides the link between the heat equation and the reflecting Brownian motion is that if u solves (1) then Let the restriction of T t to U be denoted byT t . Define The following argument was introduced in [1] (in a slightly different context). As a result of Corollary 2.1 and equation (10), we have that Note also that the spectral radius ofT t is e −µ 2 t .
We recall an abstract result on linear operators that leave a cone invariant. Let E be a real normed space. A closed set K ⊂ E is called a cone if for all a, b ∈ K, α ≥ 0 one has a+b, αa ∈ K, and if K ∩ −K contains only the zero vector. If a, b ∈ E and b − a ∈ K, one writes a ≤ b. The following is from [12] Theorem 9.2.

Theorem 3.2 Let
A be an operator in a real Banach space E that leaves a cone K invariant (i.e., AK ⊂ K). Assume ii. A is compact and its spectral radius satisfies r(A) > 0.

Proof of Theorem 3.1:
We verify the hypotheses of Theorem 3.2. The fact that S ∩ −S = {0} follows easily from the assumption that V has a nonempty interior. Hence S is obviously a cone in the Banach space U with the norm of L 2 (D). To show that S − S = U it is enough to show that every u ∈ U that satisfies a global Lipschitz condition can be written as the difference between two elements of S. Assume then that By Condition 2.3 there must be γ and > 0 such that v, γ ≥ |v| for all v ∈ V . Let x 0 be such for all x, y ∈ D, x − y ∈ V . That is, u + κ −1 z ∈ S. Obviously z ∈ S, hence we obtain that S − S = U . As discussed before,T t leaves S invariant. Moreover, r(T t ) = e −µ 2 t > 0. Hence by Theorem 3.2 there is an eigenfunction φ 2 in S, corresponding to µ 2 .
To conclude (9) we proceed as in the proof of Theorem 3.3 of [1]. Every eigenfunction must be real analytic in D [1] hence cannot be constant in an open set unless it is identically zero. However, if y is any interior point then φ 2 (y) ≤ φ 2 (x) if x belongs to the set (y + V ) ∩ D, which has non-empty interior. Consequently, φ 2 cannot attain its maximum at y.

Simplicity in dimension two
In this section we deal solely with planar domains. It is known since [16] that the multiplicity of µ 2 for simply connected planar domains is at most two. For a certain family of planar domains we prove that µ 2 is simple.

Theorem 4.1 Let D be a bounded domain of the form
where g and h are non-decreasing C 2 functions, g is convex and h concave. Let (ξ 1 , g(ξ 1 )) and (ξ 2 , g(ξ 2 )) denote the two corner points (i.e., ξ 1 and ξ 2 are the only two solutions ξ to the equation g(ξ) = h(ξ)). Assume that for i = 1, 2, both g and h are affine at some neighborhood of ξ i and the angle between the graphs of g and h near ξ i is greater than or equal to π/4. Then the second Neumann eigenvalue on D is simple.
Before proving the above result, let us show that Theorem 3.1 can be applied to domains in R 2 to recover the following result of [1], section 3, specialized to domains with convex corners.

Then there is an eigenfunction corresponding to the second Neumann eigenvalue on D such that for all y ∈ D inf
Proof: We will show that the assumptions of Then it is easy to see that the set {x : 0 ≤ x 2 ≤ max(m g , m h )x 1 } is an invariant wedge for D as well.
We introduce some notation. Consider the Banach spaceŨ of functions on D that satisfy a global Lipschitz condition as well as For i = 1, 2, let Ξ i denote the corner point (ξ i , g(ξ i )). Let also Recall that by assumption g and h are affine near the corner points and let r be some fixed (small enough) number such that B r (Ξ i ) ∩ D, i = 1, 2 are sectors. Set In what follows, c denotes a positive constant whose value may change from line to line. We state three lemmas and prove them in the end of this section.  Let u(t, x) be the solution to the heat equation (1) with initial condition u 0 ∈ L 2 (D) and Neumann boundary conditions. Then u (1, x) is globally Lipschitz in D.

Lemma 4.2 Let the assumptions of Theorem 4.1 hold. If φ ∈S is an eigenfunction, then φ ∈ S .
Recall that P x,y denotes the probability law under which (X, Y ) is a two point reflecting Brownian motion in D started at (x, y).

Lemma 4.3 Let the assumptions of Theorem 4.1 hold. Let B be some disc in D, away from its boundary. Then there is a c = c(B) > 0 such that for all small enough, the estimate
We use several times the fact that the process |X t − Y t | is nonincreasing in t, P x,y -a.s. To see this, use (4) and the notation of Section 2 to write For any convex domain it holds that when x s ∈ ∂D, δ s , ν x s ≤ 0 (recall that δ s = x s − y s and ν x s is an inward normal to ∂D at x s ). The first integral on the right hand side of the last display is therefore nonincreasing in t. A similar argument reveals that the second integral is nondecreasing, and it follows that |δ t | is nonincreasing in t. Therefore |X t − Y t | is a.s. nonincreasing.

Proof of Theorem 4.1:
We assume that µ 2 is not simple and argue by contradiction.
By Theorem 4.2 there is an eigenfunction φ corresponding to µ 2 with φ ∈ S (and where V = R 2 + ). Let φ ⊥ be an eigenfunction in the second eigenspace, orthogonal to φ. Both φ and φ ⊥ are assumed to have unit L 2 norm. By Lemma 4.1, both φ and φ ⊥ are inŨ . It follows that φ ∈S. Moreover, since φ ⊥ and −φ ⊥ cannot simultaneously belong toS, we assume (without loss) that and let a * = inf{a ∈ [0, 1] : φ a ∈S}. Let also ψ = φ a * . Note that φ a is continuous as a mapping from [0, 1] toŨ , and thatS is closed inŨ . Therefore the set of a ∈ [0, 1] for which φ a ∈S is closed, and since this set does not contain 1, it follows that a * < 1. Furthermore, we have that ψ ∈S and ψ = 0. Define and let K be its limit as a ↓ a * in the following sense: Since a * < 1, there must exist a sequence a n ↓ a * such that K an are non-empty. Hence Since ψ ∈S, by Lemma 4.2 ψ ∈ S . We claim that for any compact A ⊂ D, To see this, note that one can write and (13) follows. Hence K must be a subset of the boundary. Let = (a) = inf{ 1 : Since K is a subset of the boundary, (a) → 0 as a ↓ a * . Moreover, on a subsequence of a ↓ a * one must have (a) > 0. Therefore, on this subsequence, one must have that M (φ a ; x) ≤ 0 for some x ∈ D . Nevertheless, we claim the following.
This claim stands in contradiction with the preceding paragraph. Therefore, once it is proved, we may infer that there can be no two orthogonal eigenfunctions corresponding to µ 2 . In what follows we prove the claim.
We estimate the two terms above as follows. First, by the monotonicity property Therefore It is well known that the density of X 1 started at x is bounded above and below by positive constants that do not depend on x ∈D. We get Note that by (14), the (possibly negative) quantity inf D M (φ a ) approaches zero as a ↓ a * .
Let B be some disc in D, away from the boundary. The estimate on the second term of (16) is obtained separately for x, y ∈ D \ B r/2 (Ξ) and for x, y ∈ D ∩ B r (Ξ). The former case is treated as follows. Recall that on D , M (φ a ) ≥ 0 and that |X s − Y s | ≤ |x − y| ≤ /4, s > 0. If X 1 ∈ D 2 then Y 1 ∈ D and therefore Z a 1 1 X 1 ∈D 2 is a.s. nonnegative. It follows that On the event in the above indicator, both X s and Y s are in D for s ∈ [0, 1] and therefore It follows that the last display is where B is a disc concentered with B and of radius rad(B) + /4. The factor in square brackets is ≥ c for small. This is a consequence of the relation between the density of a Brownian motion killed at the boundary and the Dirichlet problem and e.g., Theorem 4.2.5 with Lemma 4.6.1 in Davies [7], that together establish a lower bound on the density of the order of the distance dist(x, ∂D) to the boundary (recall that the boundary is C 2 away from the corners). Hence by (13), In fact, (18) holds for x, y ∈ B r (Ξ) ∩ D as well. Indeed, combining the fact that Z a 1 1 X 1 ∈D 2 ≥ 0 with (13) and Lemma 4.3, As noted before, inf D M (φ a ) approaches zero as a ↓ a * . Hence for all a − a * small the right hand side must be positive. The claim is therefore proved. This concludes the proof of the theorem.
Proof of Lemma 4.1: Consider the Sobolev space W 1,2 (D) with the norm where · is the norm in L 2 (D). It is well known (e.g. from [13] Theorem III.5.1) that for domains with Lipschitz boundary, T 1/2 maps L 2 (D) into W 1,2 (D). Hence it suffices to show that T 1/2 u is Lipschitz whenever u ∈ W 1,2 (D). Fix u ∈ W 1,2 (D) and let u k be a sequence of globally Lipschitz functions on D converging to u in W 1,2 (D). Assume without loss that u k ≤ 2 u . We show below that for each k, T 1/2 u k is globally Lipschitz with constant c(D) u k , where c(D) depends only on D. Since T t is continuous on W 1,2 and the set of functions with Lipschitz constant 2c(D) u is closed in W 1,2 , T 1/2 u is Lipschitz, and the result follows.
Indeed, let y and x j , j = 1, 2, . . . be in D such that x j converge to y and x j = y, j = 1, 2, . . .. Consider the probability space (Ω, F, (F t ), P ) of Section 2 and the F t -Brownian motion W . Let E denote expectation with respect to P . For each j let X j denote the solution to the Skorokhod problem (x j + W, D, n) and as before let Y denote the solution to (y + W, D, n). Recall that the processes just defined satisfy (8) with the corresponding initial conditions, hence (10) is applicable to each of them. Using also the fact that |X t − Y t | is nonicreasing a.s. it follows that for each j (we have k and t fixed in what follows) Using a.s. monotonicity again, X j t converges to Y t a.s. and therefore a.s., However, u k is Lipschitz hence Z j are uniformly bounded. We therefore apply Fatou's lemma and have lim sup where the last inequality follows from the well known fact that the density of Y t is uniformly bounded for y ∈ D (and t fixed) (see e.g. [1] p. 6).

Proof of Lemma 4.2:
Recall that any eigenfunction is real analytic in D (cf. [1]). Let φ ∈S be an eigenfunction. We first show there must be a disc B where M (φ) ≥ c B > 0. Assume this is not the case. Then for all x ∈ D, ∇φ(x), e i = 0 for either i = 1 or i = 2. Then either there is a disc in D where ∇φ(x) = |∇φ(x)|e 1 , or there is a disc in D where ∇φ(x) = |∇φ(x)|e 2 . Assume then that B is a disc where ∇φ(x) = |∇φ(x)|e 1 (the other case is treated similarly). Let A be a (relatively open) subset of the boundary ∂D where the inward normal n satisfies n(z), e 1 > 0 and n(z), e 2 < 0, for z ∈ A. (It is obvious that there must exist such A.) Now let x, y ∈ B be such that x − y = αe 2 , where α > 0. Recall that under P x,y , (X, Y ) denotes a two-point reflecting Brownian motion inD, started at (x, y). Consider the event η that (1) X hits ∂D at A before time 1 without hitting ∂D outside A till time 1; (2) |L x | 1 > 0; (3) Y never hits ∂D before time 1; and (4) X 1 , Y 1 ∈ B . This event has positive P x,y -probability, as can be seen e.g., as follows. First, construct a C 1 path w . , for which the solutionsx . ,ỹ . to the Skorokhod problems (x + w . , D, n) and respectively, (y + w . , D, n) satisfy the conditions (1)-(4) above. Then recall that for convex domains the Skorokhod map w . →x . is continuous in the uniform topology (see [14], Theorem 1.1). Therefore these conditions are also satisfied when w . is replaced by any element of the tube provided δ is small enough. But the Wiener measure assigns positive probability to such tubes. Therefore η has positive probability. Now, recall that Hence on η one has that X 1 − Y 1 , e 1 > 0. Thus Since . This contradicts the assumption that ∇φ = |∇φ|e 1 in B , and therefore there must exist a disc B where M (φ) ≥ c B > 0. Let such a disc B be fixed.
Consider now a disc B 1 = B ρ (z), where z ∈ D. Take any x, y ∈ B 1 with x − y ∈ V . Consider the event that dist(X t , ∂D) > 2ρ, t ∈ [0, 1] and that X 1 , Y 1 ∈ B. Note that on this event Y never hits ∂D before time 1. Also, for ρ small, this event has positive P x,y -probability that moreover, is bounded below by a positive constant c that does not depend on x, y (satisfying the condition just stated). Hence T 1 φ(x) − T 1 φ(y) ≥ cc B |x − y| for all such x, y. This implies that M (φ; x) ≥ cc B e µ > 0, x ∈ B ρ (z), where µ is the corresponding eigenvalue. Since z ∈ D is arbitrary, φ ∈ S . This concludes the proof of the lemma.
Consider Ξ 1 as the origin. Let B 1 be some fixed ball in the sector σ θ 1 ,θ 2 r , away from the boundary of D. Note first that one can replace the expectation of |X 1 − Y 1 |1 X 1 ∈B by that of |X 1/2 − Y 1/2 |1 X 1/2 ∈B 1 at the cost of a constant, since  where N i,j,k = {n ∈ R 3 : |n| = 1, n, p i ≤ n, p j = 0 ≤ n, p k }.
Let V = cone{v ∈ R 3 : |v| = 1, |v − p i | ≤ a, i = 1, 2, 3}. Figure 1(a) sketches the set V and several subspaces orthogonal to elements of N , intersected with the unit sphere. For example, n and t represent a normal in N 2,1,3 and its orthogonal subspace. As n varies within N 2,1,3 , t varies over the planes that pass through 0 and p 1 and between p 2 and p 3 . Proof: The elementary details are omitted. Let v satisfy |v| = 1, |v − p i | ≤ a, i = 1, 2, 3 and let n be in N 1,2,3 . We will first show that π n v ∈ V . Let m 1,2 be a normal to the plane generated by p 1 and p 2 and such that m 1,2 , p 3 > 0. Define similarly m 2,3 with the condition m 2,3 , p 1 > 0. Then it is easy to show that the following condition is sufficient: (1) z = πnv |πnv| − p 2 satisfies |z − p 2 | ≤ a, (2) m 1,2 , z ≥ 0, and (3) m 2,3 , z ≥ 0. This condition can be verified by direct calculation. A similar argument holds for the other sets N i,j,k .
The symmetry of the (p i ) in the last example is not necessary and we have assumed it only for the ease of presentation. We describe more examples without proof. Figure 1(b) shows a set of tangential subspaces and a corresponding invariant wedge. The subspaces all pass through either q 1 or q 2 . Figure 2(a) depicts a different structure. Figure 2(b) shows a continuous version of this structure. In particular, any tangent plane to ∂D (at a smooth portion) will be parallel to a plane that is tangential to the curve c and passes though the origin.
In the rest of this section, we prove a result needed in the proof of Lemma 2.1.

Lemma 5.1
Let V ⊂ R d be a convex closed set and for > 0 let V be defined as in (5). Then the function u ∈ V c → m ∈ S d−1 defined as the unique unit inward normal to ∂V at u, where = dist(u, V ), is continuous. Let now u ∈ V c and σ > 0 be given. We will show that there exists an η > 0 such that u ∈ V c and |u − u | < η imply |m(u) − m(u )| < σ. Let = dist(u, V ) and v ∈ ∂V be such that |u − v| = . For u ∈ V c let v ∈ ∂V be such that |u − v | = dist(u , V ). Choose α such that u = u + αm(u ) will satisfy dist(ũ, V ) = . Note that dist(ũ, V ) = |ũ − v | and as a consequence m(ũ) = m(u ).
We claim that the following inequality must hold |v − v | ≤ |u −ũ|.