Continuous Maps from Spheres Converging to Boundaries of Convex Hulls
CCONTINUOUS MAPS FROM SPHERES CONVERGINGTO BOUNDARIES OF CONVEX HULLS
J. MALKOUN AND P.J. OLVER
Abstract.
Given n distinct points x , . . . , x n in R d , let K denotetheir convex hull, which we assume to be d -dimensional, and B = ∂K its ( d − f ε : S d − → K which, for ε >
0, are defined on the ( d − f ε ( S d − ) are codimension 1 submanifoldscontained in the interior of K . Moreover, as the parameter ε goes to0 + , the images f ε ( S d − ) converge, as sets, to the boundary B of theconvex hull. We prove this theorem using techniques from convexgeometry of (spherical) polytopes and set-valued homology. Wefurther establish an interesting relationship with the Gauss mapof the polytope B , appropriately defined. Several computer plotsillustrating our results will be presented.
1. Introduction
Given a configuration X = ( x , . . . , x n ) of n distinct points in R d ,computing their convex hull K = Conv( X ) is a famous problem inComputational Geometry. Many algorithms have been developed forthis task, including the Gift Wrap or Jarvis March algorithm, the Gra-ham Scan algorithm, QuickHull, Divide and Conquer, Monotone Chainor Andrew’s algorithm, Chan’s algorithm, the Incremental Convex Hullalgorithm, the Ultimate Planar Convex Hull algorithm, and others.See, for instance, [3] and the references within.In this paper, we develop an alternative, direct approach to this prob-lem that does not rely on any underlying computer algorithm. Instead,assuming dim K = d , meaning that its interior K ◦ is a nonempty opensubset of R d , we construct a one-parameter family of approximations toits ( d − B = ∂K , as the images of continuousmaps f ε : S d − → R d for ε >
0, that are defined explicitly, and fairlysimply, in terms of the points x , . . . , x n .Initial computer generated plots suggested that the images f ε ( S d − )of our family of maps provide excellent approximations to the boundary Date : July 8, 2020. a r X i v : . [ m a t h . M G ] J un J. MALKOUN AND P.J. OLVER B for all configurations that we have tried; see Figures 1 and 2 forsome representative examples. Our main result, Theorem 2.1, statesthat the images f ε ( S d − ) converge, as sets, to the boundary B as theparameter ε → + . We will also explain in detail the mechanism ofconvergence. We then establish a relationship with the Gauss map ofa smooth surface, [7], thereby defining the inverse Gauss map of theboundary of the convex hull as a set-valued map. Indeed, our proofof the main theorem relies on techniques from the theory of set-valuedhomology.On the other hand, the convergence of the approximating sets f ε ( S d − )to the boundary B is highly non-uniform. Indeed, as we will see, the im-ages f ε ( n ) of almost every point n ∈ S d − converge to one of the verticesof B . Thus, if one discretely samples S d − by a large but finite num-ber of points y , . . . , y N , most of their image points f ε ( y k ) ∈ f ε ( S d − )will accumulate around the vertices of B , and the remainder of B willbe increasingly sparsely approximated as ε → + . This non-uniformsampling property can be observed in the three-dimensional illustrativeplots in Figure 2.See [6] for an alternative, less practical approach to approximatingconvex polytopes and convex sets by algebraic sets. A potential futureproject based on these constructions will be to develop fast practicalalgorithms for approximating the convex hull of a point configuration.A potentially interesting extension of our techniques will be to theapproximation of Wulff shapes of crystals, [11].
2. A Family of Maps defined by a Point Configuration
Let us begin by introducing the basic set up and our notation, beforedefining the family of maps that will be our primary object of study.Let C n ( R d ) denote the configuration space of n distinct points in R d .Let X = ( x , . . . , x n ) ∈ C n ( R d ), so that each x i ∈ R d and x i (cid:54) = x j whenever i (cid:54) = j . Assuming n ≥ d + 1, let C ∗ n ( R d ) ⊂ C n ( R d ) denotethe dense open subset of nondegenerate configurations , meaning thosewhose points do not all lie on a proper affine subspace of R d . Fromhere on we fix the nondegenerate point configuration X ∈ C ∗ n ( R d ), andsuppress all dependencies thereon.Let K = Conv( X ) ⊂ R d denote the convex hull of the points in X ,which, by nondegeneracy, is a bounded convex polytope of dimension d whose interior is a nonempty open subset K ◦ ⊂ R d , [8, 15]. Let B = ∂K = ∂ Conv( X ) be its boundary, which is a piecewise linearclosed hypersurface in R d , forming a ( d − APS CONVERGING TO CONVEX HULLS 3
Given any pair of indices 1 ≤ i, j ≤ n with i (cid:54) = j , we define real-valued functions c ij : R + × S d − → R + = { < t ∈ R } by c ij ( ε, n ) = ε + max { , − (cid:104) n , n ij (cid:105)} , ε > , n ∈ S d − , (2.1)where (cid:104) · , · (cid:105) denotes the Euclidean inner product in R d , and where n ij = x j − x i (cid:107) x j − x i (cid:107) ∈ S d − , i (cid:54) = j, (2.2)is the unit vector pointing from x i to x j , with (cid:107) · (cid:107) denoting the Eu-clidean norm. Note that n ij = − n ji . The c ij in (2.1) are con-tinuous maps; moreover, c ij ( ε, n ) > ε >
0. We further define, for any 1 ≤ i ≤ n , the map c i : R + × S d − → R + by the ( n − c i ( ε, n ) = (cid:89) ≤ j ≤ nj (cid:54) = i c ij ( ε, n ) . (2.3)Finally, let us set λ i ( ε, n ) = c i ( ε, n )∆( ε, n ) , i = 1 , . . . , n, (2.4)where∆( ε, n ) = n (cid:88) j =1 c j ( ε, n ) > ε > , n ∈ S d − . (2.5)Thus, 0 < λ i ( ε, n ) < , and n (cid:88) i = 1 λ i ( ε, n ) = 1 . (2.6)Given a point configuration X ∈ C n ( R d ), we can now define themain object of interest in this paper: the one-parameter family of maps f ε : S d − → R d defined by f ε ( n ) = n (cid:88) i = 1 λ i ( ε, n ) x i , ε > , n ∈ S d − . (2.7)From (2.6), (2.7), one immediately deduces that f ε ( n ) ∈ K ◦ , for any ε > , n ∈ S d − . Inspection of Figures 1 and 2, and others that can be easily generatedby computer, indicates that, for a given X ∈ C ∗ n ( R d ) and small ε > S d − under f ε may be used as a good approximation ofthe boundary B = ∂K ⊂ R d of the convex hull of X . More precisely,the Main Theorem to be proved in this paper is as follows: J. MALKOUN AND P.J. OLVER (a) n = 3 (b) n = 4 (c) n = 5(d) n = 7 (e) n = 10 (f) n = 15 Figure 1.
Plots of planar point configurations X andthe image f ε ( S ), with ε = 0 . Theorem 2.1.
Given X ∈ C ∗ n ( R d ) , let K = Conv( X ) be their convexhull, which has dimension d . Let f ε be defined by (2.7) . Then, for ε > ,the images of the unit sphere under f ε lie in the interior of the convexhull of X , so f ε ( S d − ) ⊂ K ◦ , and, moreover, converge to its boundaryas sets in R d : lim ε → + f ε ( S d − ) = ∂K. (2.8)The set theoretic convergence in (2.8) is uniform in the sense thatthe images f ε ( S d − ) lie in an O( ε ) neighborhood of the boundary ∂K ,even though their pointwise convergence is highly nonuniform. Seebelow for precise details on what this means. Remark : On the other hand, if the point configuration is degenerate,meaning X ∈ C n ( R d ) \ C ∗ n ( R d ) and so its convex hull K = Conv( X )has dimension < d , then one can show that lim ε → + f ε ( S d − ) = K .Indeed, observe that the maps f ε depend continuously on the pointconfiguration. If one slightly perturbs X to a nondegenerate configura-tion X δ ∈ C ∗ n ( R d ), then their perturbed convex hull K δ is of dimension APS CONVERGING TO CONVEX HULLS 5 (a) X consists of the vertices of a regular tetrahedron, so n = 4.(b) X consists of the vertices of a cube, so n = 8. Figure 2.
Plots of point configurations X in dimension d = 3 and the images of sample points on S under themap f ε with ε = 0 . d and, by the Theorem, f ε ( S d − ) → ∂K δ . But as δ →
0, their bound-aries converge to the entire convex hull: ∂K δ → K , which enables oneto establish the result. Since this case is of less importance for ourpurposes, the details are left to the reader. J. MALKOUN AND P.J. OLVER
Figure 3. X consists of the vertices of a triangle, with ε = 0 .
1. One can see that f ε ( S ) is indented at the points f ε ( n ), shown as small red rectangles, which correspondto n ∈ S ij , for 1 ≤ i, j ≤ i (cid:54) = j .In Section 3, we present notions from convex geometry that are rel-evant to this work, including normal cones and normal spherical poly-topes. The latter enable us to associate with a convex polytope B aspherical complex S ∗ B , [10], meaning a tiling of S d − by spherical poly-topes, with the property that it has the same combinatorial type as thedual polytope B ∗ . Then, in Section 4, we connect our constructionswith the differential geometric concept of the Gauss map of a convexhypersurface, generalized to the boundary of the convex polytope. Weexplain how our maps converge to the inverse Gauss map of the bound-ary of the convex hull of the point configuration, which is viewed as aset-valued function. Finally in Section 6, we prove our main result us-ing a combination of convex geometry and set-valued homology theory,the latter described in Appendix A.
3. Convex Geometry and (Spherical) Polytopes
Let us recall some basic terminology and facts about convex setsand cones, and both flat and spherical polytopes, many of which canbe found in [4, 14]. The closed cones appearing in this paper are convex, pointed , meaning they do not contain any positive dimensional linearsubspace of R d , and polyhedral , meaning they can be characterized asthe intersection of finitely many, and at least two, closed half spaces,[5, 15]. On the other hand, for us an open cone N ⊂ R d is a cone suchthat N \ { } is an open subset of R d and such that its closure N is ofthe above type.Let us fix a nondegenerate point configuration X ∈ C ∗ n ( R d ) consist-ing of n distinct points x , . . . , x n ∈ R d . Let K = Conv( X ) ⊂ R d APS CONVERGING TO CONVEX HULLS 7 denote the convex hull of the points in X , which is a bounded con-vex d -dimensional polytope, [8, 15]. Let B = ∂K = ∂ Conv( X ) be itsboundary, which is itself a polytope of dimension d − R d . Assume, by relabelling ifnecessary, that x , . . . , x κ are the vertices of K , while x κ +1 , . . . , x n arethe remaining points, which may either lie in the interior K ◦ or at anon-vertex point of the boundary B . The faces of B range in dimen-sion from 0, the vertices , to 1, the edges , up to d −
1, the facets . Twovertices are adjacent if they are the endpoints of a common edge. Notethat each face F ⊂ K is itself a convex polytope. If 0 < m ≤ d − interior of an m -dimensional face F by F ◦ = F \ ∂F ,which is a flat m -dimensional submanifold of R d . (Keep in mind thatthis is not the same as its interior as a subset of R d , which is empty.)Define the normal cone at the point x i by N i = (cid:8) y ∈ R d (cid:12)(cid:12) (cid:104) y , n ij (cid:105) ≤ j (cid:54) = i (cid:9) = (cid:92) j (cid:54) = i H ij , (3.1)where the unit vectors n ij ∈ S d − are given in (2.2), and H ij = (cid:8) y ∈ R d (cid:12)(cid:12) (cid:104) y , n ij (cid:105) ≤ (cid:9) (3.2)is the closed half space opposite to n ij . Further let N ◦ i = (cid:8) y ∈ R d (cid:12)(cid:12) (cid:104) y , n ij (cid:105) < j (cid:54) = i (cid:9) . (3.3)denote the interior of the normal cone N i . It is easy to see that N ◦ i (cid:54) = ∅ if and only if x i is a vertex. Also, N ◦ i ∩ N ◦ j = ∅ whenever i (cid:54) = j . Indeed,if y ∈ N ◦ i then (cid:104) y , n ij (cid:105) <
0. But then (cid:104) y , n ji (cid:105) = (cid:104) y , − n ij (cid:105) > y (cid:54)∈ N ◦ j . Furthermore, the union of the vertex normal cones isthe entire space: κ (cid:91) i =1 N i = R d , (3.4)i.e., every vector is in one of the normal cones. This is a direct conse-quence of the Supporting Hyperplane Theorem; see for instance [4, pp.50–51].A spherical polytope is characterized as the intersection of finitelymany closed hemispheres that does not contain any antipodal points,cf. [5, § C ∩ S d − of the unit sphere with a pointed polyhedral cone C ⊂ R d .Let us consequently define the normal spherical polytope S i = N i ∩ S d − = (cid:8) n ∈ S d − (cid:12)(cid:12) (cid:104) n , n ij (cid:105) ≤ j (cid:54) = i (cid:9) , (3.5) J. MALKOUN AND P.J. OLVER associated with the point x i . Its interior S ◦ i = N ◦ i ∩ S d − = (cid:8) n ∈ S d − (cid:12)(cid:12) (cid:104) n , n ij (cid:105) < j (cid:54) = i (cid:9) (3.6)is nonempty if and only if x i is a vertex, in which case it is an opensubmanifold of the unit sphere. Note that, by (3.4) and the precedingremarks, κ (cid:91) i =1 S i = S d − , S ◦ i ∩ S ◦ j = ∅ , i (cid:54) = j. (3.7)The normal cone and normal spherical polytope associated with ageneral point x ∈ K in the convex hull are similarly defined: N x = (cid:8) y ∈ R d (cid:12)(cid:12) (cid:104) y , z − x (cid:105) ≤ z ∈ K (cid:9) ,S x = N x ∩ S d − = (cid:8) n ∈ S d − (cid:12)(cid:12) (cid:104) n , z − x (cid:105) ≤ z ∈ K (cid:9) . (3.8)As above, N x = { } if x ∈ K ◦ , while N x i = N i when x i is a vertex.More generally, if F ⊂ B = ∂K is an m -dimensional face, then thenormal cone N x is independent of the point x ∈ F ◦ in its interior,and we thus define N F = N x for any such x ∈ F ◦ . If the face F hasdimension m , then N F is a ( d − m )-dimensional cone. Define its interior to be N ◦ F = N F \ ∂N F , which is a ( d − m )-dimensional submanifoldof R d . Warning : unless m = 0, so F is a vertex, N ◦ F is not the sameas the interior of N F considered as a subset of R d , which is empty. Inparticular, if F is a facet, i.e., a ( d − N F isa one-dimensional cone, i.e., a ray in the direction of its unit outwardnormal n F , with N ◦ F = { c n F | c > } . Observe that if H ⊂ ∂F isa subface, then N F ⊂ ∂N H . Further, convexity of K implies that N ◦ F ∩ N ◦ G = ∅ whenever F (cid:54) = G are distinct faces of B ; in particular, n F (cid:54) = n G whenever F (cid:54) = G are distinct facets.The collection of the interiors of all the normal cones to the faces of B form the complete normal fan associated with the polytopes B and K , and their disjoint union fills out the entire space, except for theorigin (which can be identified with N K ): R d = { } ∨ (cid:95) F ⊂ B N ◦ F . (3.9)We further define the normal spherical polytope associated with the m -dimensional face F as S F = N F ∩ S d − . When m < d −
1, itsinterior S ◦ F = N ◦ F ∩ S d − is a ( d − m − S d − , while for m = d −
1, the normal spherical polytope S F is a singlepoint, namely the facet’s unit outward normal n F . As an immediateconsequence of the complete normal fan decomposition (3.9), we can APS CONVERGING TO CONVEX HULLS 9 write the sphere as a disjoint union S d − = (cid:95) dim F 1. They associate each facet F ⊂ B with itsoutward normal n F ∈ S d − ⊂ R d . The outer normal transform of B is defined to be the convex hull of the facet normals in R d . They ob-serve that, unlike our spherical dual, their transform is not necessarilycombinatorially equivalent to the dual polytope B ∗ .On the other hand, if we flatten all the normal spherical polytopes ofthe spherical dual S ∗ B , meaning we replace each S F ⊂ R d by the convexhull of its vertices, the result will be a polytope (cid:98) B ⊂ R d containedwithin the unit ball, all of whose vertices lie on the unit sphere. Al-though the resulting polytope (cid:98) B also has the same combinatorial typeas B ∗ , it is not necessarily convex. The outer normal transform of B can thus be identified with the convex hull of (cid:98) B , and so, when (cid:98) B is notconvex, will possess a different combinatorial structure than B ∗ .Indeed, counterexamples to the problem of inscribing convex poly-topes of a given combinatorial type in spheres, [13], are of this form. Forexample, the dual to the truncated tetrahedron, known as the triakistetrahedron, is not inscribable in a sphere. The flattened version of thespherical dual to a truncated tetrahedron is a cube with diagonals thatbisect each square into a pair of triangular facets, and form the edgesof an interior tetrahedron. Both the spherical dual and the resultingflattened cube with diagonals have the same combinatorial type as thetriakis tetrahedron. However, the flattened cube, while inscribed in theunit sphere, is not a convex polyhedron since it has pairs of coplanartriangular facets possessing a common normal. It is, of course, the set-theoretic boundary of a convex subset of R , namely the inscribed solidcube, whose cubical boundary (without the diagonals) can be identifiedas the outer normal transform of the original truncated tetrahedron, and is not combinatorially equivalent to the triakis tetrahedron. Fur-thermore, slightly perturbing the original truncated tetrahedron leadsto a perturbed spherical dual and a perturbed cube with diagonals thatis inscribed in the sphere, again both having the same combinatorialtype as the triakis tetrahedron. However, although its triangular facesare no longer coplanar, the resulting polyhedron is not the boundary ofa convex subset of R , and hence not equal to its outer normal trans-form, which is the convex hull of this nonconvex perturbed cube. Allthis is a necessary consequence of the non-inscribability of the triakistetrahedron.In general, if the flattened spherical dual of a polytope is convexthen it has to coincide with its outer normal transform, which is then,by the above remarks, combinatorially equivalent to the dual polytope.On the other hand, if it is not convex then its convexification, whichis the outer normal transform, cannot be combinatorially equivalent tothe dual. Thus, we have established the following result. Proposition 3.1. Let B ⊂ R d be a convex polytope of dimension d − .Then the outer normal transform of B is combinatorially equivalent tothe dual polytope B ∗ if and only if the flattened spherical dual of B isconvex. Finally, for later purposes, we will introduce some useful open subsetsof the normal spherical complex (3.10). If F ⊂ B is a facet withoutwards unit normal n F ∈ S d − , so S F = { n F } , set W F = S F ∨ (cid:95) G (cid:40) F S ◦ G (3.11)where the union is over the proper subfaces G (cid:40) F . On the other hand,if F ⊂ B is a face with 1 ≤ dim F < d − 1, set W F = (cid:95) G ⊆ F S ◦ G . (3.12) Lemma 3.2. Under the above definitions, W F is a relatively open sub-set of S d − .Proof : This follows from the fact that the corresponding union of nor-mal cones V F = { } ∨ (cid:95) G ⊆ F N ◦ G is an open cone and W F = V F ∩ S d − . Indeed, one can use a per-turbed version of the Supporting Hyperplane Theorem that says thata perturbed supporting hyperplane remains supporting at some pointin its support. In more detail, if H is a supporting hyperplane such APS CONVERGING TO CONVEX HULLS 11 that H ∩ B = F where F is a face, and (cid:101) H is any sufficiently smallperturbation of H , then (cid:101) H ∩ B = G for some subface G ⊆ F . Keepin mind that the subface could be a vertex. Q.E.D. The last result of this section is a technical construction, that is keyto our proof of the Main Theorem 2.1. The reader may wish to skip itfor now, and return once the proof is underway. Proposition 3.3. Let F ⊂ B be a face of dimension ≤ m ≤ d − .Let S F be its normal spherical polytope and W F ⊂ S d − the open subsetgiven by Lemma 3.2. Let G , . . . , G k be its ( m − -dimensional sub-faces, so that ∂F = (cid:83) ki =1 G i . Similarly, let S ◦ i = S ◦ G i and W i = W G i .Suppose N ⊂ W F ⊂ S d − be a connected m -dimensional submanifoldsuch that either ( a ) if F is a facet, of dimension d − , with unitoutwards normal n = n F , then N is an open neighborhood of n , or ( b ) if ≤ m = dim F < d − , then N intersects S ◦ F transversally at asingle point n ∈ N ∩ S ◦ F . Then if (cid:101) N ⊂ N is a sufficiently small opencontractible submanifold with n ∈ (cid:101) N , which implies n ∈ ∂ ( (cid:101) N ∩ S ◦ i ) for all i = 1 , . . . , k , we can decompose its boundary ∂ (cid:101) N = (cid:83) ki =1 L i intothe union of ( m − -dimensional submanifolds that only overlap ontheir boundaries, meaning L i ∩ L j = ∂L i ∩ ∂L j whenever i (cid:54) = j , withthe property that each L i ⊂ W i intersects S ◦ i transversally at a singlepoint n i ∈ L i ∩ S ◦ i = ∂ (cid:101) N ∩ S ◦ i .Proof : Choose r > N r = { n ∈ N | (cid:107) n − n (cid:107) < r } has boundary ∂N r = { n ∈ N | (cid:107) n − n (cid:107) = r } . Moreover, reducing r if necessary, we claim that ∂N r intersects each S ◦ i transversally at a single point n i ∈ ∂N r ∩ S ◦ i .Indeed, in a small neighborhood n ∈ U we can choose local coor-dinates centered at n such that, locally, S ◦ F ∩ U is a ( d − m − N r ⊂ U is a transverse m -dimensional subspace,while S ◦ i ∩ U is a ( d − m )-dimensional half space with local boundary ∂S i ∩ U = S ◦ F ∩ U , from which the preceding claim is evident.We now set (cid:101) N = N r . Since S F ⊂ ∂S i , this immediately implies n ∈ ∂ ( (cid:101) N ∩ S ◦ i ). The final task is to decompose ∂ (cid:101) N = (cid:83) ki =1 L i asin the statement of the Proposition. It is reasonably clear that thereare many ways to do this, but for definiteness here is one possibleconstruction. First we note that, by (3.11), (3.12), either W F \ { n F } = k (cid:91) i =1 W i , or W F \ S ◦ F = k (cid:91) i =1 W i , according to whether F is a facet or not. We thus, for each i = 1 , . . . , k ,need to choose L i ⊂ (cid:101) N ∩ W i with the requisite properties.First, define the closed subset L i ⊂ ∂ (cid:101) N to be the set of all n ∈ (cid:101) N ∩ W i such that if n ∈ (cid:101) N ∩ S ◦ H for some adjacent subface H (cid:40) G i ,then dist( n , n i ) ≤ dist( n , n j ) for all other adjacent ( m − G j , meaning that H (cid:40) G j . Clearly N = (cid:83) L i and, moreover, L i and L j only overlap on their common boundary, which could beeither part of a boundary of an S H or a point n ∈ N ∩ S ◦ H that isequidistant to n i and n j . We then set L i = L ◦ i to be its interior relativeto ∂ (cid:101) N . Since n i ∈ L ◦ i , transversality of ∂ (cid:101) N to S ◦ i at n i immediatelyimplies the same for the relatively open submanifold L i . We concludethat the resulting submanifolds satisfy the required conditions. Q.E.D. 4. The Gauss Map of a Convex Polytope We claim that the preceding construction can be identified with aform of the inverse of the Gauss map of the boundary of the convexhull B = ∂K . Recall, [7], that the Gauss map of a smooth closedhypersurface, i.e., a ( d − M ⊂ R d ,is γ M : M −→ S d − , γ ( y ) = n y , y ∈ M, (4.1)where n y denotes the unit outward normal to M at y . If M is con-vex, then its Gauss map is one-to-one and onto, with smooth inverse γ − M : S d − → M .We are interested in the convergence of the Gauss maps γ M ε associ-ated with a parametrized family of smooth closed convex hypersurfaces M ε , for ε > 0, that converge to the piecewise linear convex hypersur-face (polytope) B = ∂K as ε → + . Convergence of the Gauss mapswill be in the sense of set-valued functions, as we now describe.In general, a set-valued function , also known as multi-valued func-tions , from a space D to a space Y means a mapping F from D to thepower set 2 Y , i.e., the set of subsets of Y , [1, 2]. In other words, theimage of x ∈ D is a subset F ( x ) ⊂ Y . More generally, a set-valuedfunction maps subsets of its domain to subsets of its range in the evi-dent manner. We say that F has closed values if F ( x ) is a closed subsetof Y for all x ∈ D . In particular, any ordinary function f : D → Y canbe viewed as a set-valued function, with closed values, by identifyingthe image y = f ( x ) of a point x ∈ D with the singleton set { y } ⊂ Y . It may happen that the closure of L i is strictly contained in L i ; this can occurif there exist n l associated with nonadjacent faces G l that lie closer to the points n ∈ (cid:101) N ∩ S ◦ H than those in any adjacent face G i . But this does not affect theconstruction since every point in L i \ L i is contained in the boundary of some L j . APS CONVERGING TO CONVEX HULLS 13 The range R ⊂ Y of F is the union of all the images of points in itsdomain D , so R = F ( D ).Here is a simple example of convergence of set-valued functions. Example : Consider the ordinary functions f ε ( x ) = 2 π (1 − ε ) arctan xε for x ∈ R , ε > . (4.2)In the usual function-theoretic sense of convergence,lim ε → + f ε ( x ) = sign x = , x > , , x = 0 , − , x < . Thus, for almost every point x ∈ R , the value of f ε ( x ) converges toeither − R ,they converge, as sets, to the curve consisting of the union of the threeline segments { ( x, − | x ≤ } ∪ { (0 , y ) | − ≤ y ≤ } ∪ { ( x, | x ≥ } . We can interpret this curve as the graph of the set-valued function (cid:98) f : R −→ R given by (cid:98) f ( x ) = { } , x > , [ − , , x = 0 , {− } , x < . (4.3)The domain of (cid:98) f is D = R and its range is the interval R = [ − , 1] = (cid:98) f ( R ). Furthermore, the ranges R ε = f ε ( R ) = ( − ε, − ε ) of thefunctions (4.2) are open intervals that converge, as sets, to the closedinterval [ − , 1] forming the range of the limiting set-valued function.In general, given spaces D, R , let D × R denote their Cartesian prod-uct, and π D : D × R → D and π R : D × R → R the standard projec-tions. Any subset S ⊂ D × R which projects onto both D = π D ( S )and R = π R ( S ) defines a set-valued mapping F with domain D andrange R , given by F ( x ) = π R ( S ∩ π − D { x } ). Its inverse F − is also aset-valued mapping from R to D , given by F − ( y ) = π D ( S ∩ π − R { y } ),[1]. For the above example (4.3), (cid:98) f − : [ − , → R is given by (cid:98) f − ( y ) = ( −∞ , , y = − , { } , − < y < , [0 , ∞ ) , y = 1 . (4.4)Note that any ordinary function thus has a set-valued inverse.In convex analysis, the normal cone (3.8) is often viewed as a set-valued mapping, that maps a point x ∈ B = ∂K to its normal cone N x . (One can, of course, extend it to all of K but the values on theinterior K ◦ are trivial.) Here we consider our normal spherical polytopeconstruction as a set-valued mapping γ B from B to S d − , mapping apoint x ∈ B to its normal spherical polytope: γ B ( x ) = S x ⊂ S d − .Suppose that M ε for ε > M ε → B as ε → + . Then their Gauss maps γ M ε → γ B converge, in the sense of set-valued functions, to the set-valued normalspherical polytope map. We can thus identify the normal sphericalpolytope map constructed above as the Gauss map of a convex polytope,i.e., a piecewise linear convex hypersurface. As for our construction, the Main Theorem 2.1 shows that the func-tions f ε : S d − → R d converge, in the set-valued sense, to the inverseof the Gauss map γ − B : S d − → B associated with the boundary of theconvex hull. On the other hand, the f ε are certainly not inverse Gaussmaps themselves. Moreover, simple examples, e.g., that in Figure 3,show that the image f ε ( S d − ) is not in general a convex hypersurface.On the other hand, it might be worth investigating the set-theoreticconvergence of their possibly multi-valued Gauss maps.Finally, we note that the concept of continuity does not extendstraightforwardly to set-valued functions, [1]. The most important ana-log is contained in the following definition. Definition 4.2. Let D, R be topological spaces. A set-valued mapping f : D → R is called upper hemicontinuous at x ∈ D if and only if, forany open neighborhood V of the set f ( x ), there exists a neighborhood U of x such that f ( x ) ⊂ V for all x ∈ U . We say that f is upperhemicontinuous if it is upper hemicontinuous at every x ∈ D .It is straightforward to verify that Example 4.1 satisfies the upperhemicontinuity condition. Warning : A few authors, including [1], use the expression “uppersemicontinuous” instead of “upper hemicontinuous”. However the lat-ter terminology seems to be more accepted by the broader community,particularly as it is not in conflict with the notion of semicontinuity ofordinary functions. 5. Some Computational Lemmas Before launching into the proof of the Main Theorem 2.1, let us col-lect together some elementary computational lemmas for the functionsused to form the maps f ε defined in (2.7). APS CONVERGING TO CONVEX HULLS 15 Recalling (2.1) and (2.3), let us set c ij ( n ) = lim ε → + c ij ( ε, n ) = max { , −(cid:104) n , n ij (cid:105)} , n ∈ S d − . (5.1)and c i ( n ) = lim ε → + c i ( ε, n ) = (cid:89) ≤ j ≤ nj (cid:54) = i c ij ( n ) , n ∈ S d − . (5.2)We can thus write c i ( ε, n ) = c i ( n ) + ε b i ( ε, n ) , (5.3)where b i is a polynomial in ε of degree n − 2. In view of (3.6) and (3.7), c i ( n ) > n ∈ S ◦ i . Thus, c i ( n ) = 0 for all i = 1 , . . . , n ifand only if n ∈ Q = S d − \ S , where S = (cid:87) κi =1 S ◦ i is the disjoint unionof the interiors of the normal spherical polytopes associated with thevertices x , . . . , x κ of B . We have thus established the following result. Lemma 5.1. Given n ∈ S d − , either all c i ( n ) = 0 , or precisely one c i ( n ) > , and the rest are all zero. Moreover, in the latter case, x i isa vertex. Indeed, if n ∈ S ◦ i , then, referring to (2.3), (5.3), c i ( n ) > b i ( ε, n ) > ε > 0, whereas c j ( ε, n ) = ε k j a j ( ε, n ) for j (cid:54) = i, (5.4)with a j ( ε, n ) > ε > 0. Here the nonnegative integer k j denotes thenumber of points x k with k (cid:54) = i, j that satisfy the distance inequalitydist( x k , P i ( n )) ≤ dist( x j , P i ( n )) where P i ( n ) = x i + n ⊥ , with n ⊥ = { y ∈ R d | (cid:104) y , n (cid:105) = 0 } denoting the affine hyperplane orthogonal to n passing through x i . Note that there is either one or no value of j forwhich k j = 0.Let us finish this section by establishing a more detailed version ofLemma 5.1, valid for an arbitrary face F ⊂ B . Lemma 5.2. Let F ⊂ B be a face of dimension ≤ m ≤ d − ,with vertices x , . . . , x k . Suppose F contains l additional ( non-vertex ) points x k +1 , . . . , x k + l ∈ F ◦ , where l may be zero. Then, given n ∈ S ◦ F , c i ( ε, n ) = ε k + l − d i ( n ) + ε k + l p i ( ε, n ) , i = 1 , . . . , k + l,c j ( ε, n ) = ε k + l p j ( ε, n ) , j = k + l + 1 , . . . , n, (5.5) where d i ( n ) > , while p ( ε, n ) , . . . , p n ( ε, n ) are polynomials in ε .Proof : Let n ∈ S ◦ F . The proof follows from the fact that c ij ( ε, n ) = ε whenever x i , x j ∈ F , so that (cid:104) n , n ij (cid:105) = 0. On the other hand, (cid:104) n , n ij (cid:105) < x i ∈ F and x j (cid:54)∈ F , which, vice versa, implies c ji ( ε, n ) = ε . The proof is completed by recalling the definition (2.3)of c i ( ε, n ). Q.E.D. 6. Proof of the Main Theorem Now we turn to the proof of the Main Theorem 2.1, on the conver-gence, as ε → + , of the hypersurfaces f ε ( S d − ) ⊂ K ◦ ⊂ R d to theboundary of the convex hull B = ∂K of the point configuration X .If we formally set ε = 0 in the preceding definition (2.7) of the map f ε , Lemma 5.1 implieslim ε → + f ε ( n ) = (cid:26) x i , n ∈ S ◦ i , undefined , n ∈ Q = S d − \ (cid:87) κi =1 S ◦ i . (6.1)Thus, for almost every point n ∈ S d − , the images f ε ( n ) converge to oneof the vertices of the convex hull. However, as in Example 4.1, this doesnot imply that, as a set, f ε ( S d − ) converges to the set of vertices V = { x , . . . , x κ } . Our goal is to prove that the images f ε ( S d − ) converge,as sets, to the entire boundary B as ε → + . Specifically, we will show: • Given any neighborhood W ⊃ B , no matter how small, we canfind ε > f ε ( S d − ) ⊂ W for any 0 < ε < ε . • Given any x ∈ B , there exist points y ε ∈ f ε ( S d − ) for ε > y ε → x as ε → + .In the language of set-theoretic limits, [1], the first statement showsthat the outer limit of the sets f ε ( S d − ) is a subset of B . The secondstatement proves that B is a subset of their inner limit . Since the innerlimit is always a subset of the outer limit, this then implies that theinner limit and outer limit coincide and are equal to B .First, recall that, for r > 0, the r -neighborhood U r of a subset D ⊂ R d is the set of points that are a distance less than r (in the Euclideannorm) from D , i.e., U r = { x ∈ R d | dist( x, D ) < r } . In what follows,when we refer to an O( ε ) neighborhood of a set, by which we mean an ε dependent system of r -neighborhoods in which, for ε sufficiently small, r = c ε for some unspecified constant c .In order to understand our set-theoretic limit, we will investigatethe behavior of the images f ε ( A ) of certain subsets A ⊂ S d − , graduallybuilding up to the entire sphere. Let us begin with the simplest case:the images of a curve C ⊂ S d − . If C ⊂ S ◦ i is entirely contained in theinterior of the normal spherical polytope associated with a vertex x i for some 1 ≤ i ≤ k , then, by (6.1), f ε ( C ) → { x i } as ε → + .The next simplest case is when the curve C is contained in the unionof two adjacent vertex spherical polytopes. Thus, by relabeling, let APS CONVERGING TO CONVEX HULLS 17 x , x be adjacent vertices of B . Let E = { λ x + λ x | λ , λ ≥ , λ + λ = 1 } ⊂ B denote the edge connecting x to x . Suppose that its interior con-tains l ≥ x , . . . , x l +2 ∈ E ◦ , while the remaining points x l +3 , . . . , x n ∈ K \ E .We note that we can also write, redundantly, E = (cid:40) l +2 (cid:88) i = 1 λ i x i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ i ≥ , l +2 (cid:88) i = 1 λ i = 1 (cid:41) . (6.2)Let S , S ⊂ S d − be the normal spherical polytopes associated with x , x , respectively, while S E = ∂S ∩ ∂S is the normal sphericalpolytope associated with the edge E . Thus S ◦ , S ◦ are open subsets of S d − , while S ◦ E is a ( d − C ⊂ S ◦ ∪ S ◦ ∪ S ◦ E ⊂ S d − such that one endpoint of C lies in S ◦ andthe other lies in S ◦ , which, by connectivity, imply C ∩ S ◦ E (cid:54) = ∅ . Ourgoal is to prove that the image curves f ε ( C ) converge, as sets, to theedge E .Now, if n ∈ C ∩ S ◦ , Lemma 5.1 combined with equations (2.4) and(5.4) imply λ ( ε, n ) = 1 + ε q ( ε, n ) , λ j ( ε, n ) = ε q j ( ε, n ) , j = 2 , , . . . , n, (6.3)where q , . . . , q n are rational functions of ε depending continuously on n ∈ C . Thus, (6.3) re-establishes the fact that all of the points in f ε ( C ∩ S ◦ ) converge to the vertex x as ε → + . A similar statementholds for n ∈ C ∩ S ◦ : λ ( ε, n ) = 1 + ε q ( ε, n ) , λ j ( ε, n ) = ε q j ( ε, n ) , j = 1 , , . . . , n. (6.4)Finally, if n ∈ C ∩ S ◦ E , in view of (2.4), (2.5), (5.5), we have λ i ( ε, n ) = d i ( n ) + ε p i ( ε, n ) D ( n ) + ε P ( ε, n ) , i = 1 , . . . , l + 2 ,ε p i ( ε, n ) D ( n ) + ε P ( ε, n ) , i = l + 3 , . . . , n, (6.5)where D ( n ) = l +2 (cid:88) i = 1 d i ( n ) > , P ( ε, n ) = n (cid:88) i = 1 p i ( ε, n ) . Comparing with (6.3), (6.4), (6.5), we find that for any n ∈ C , l +2 (cid:88) i = 1 λ i ( ε, n ) = 1 + ε Q ( ε, n ) , f ε ( n ) = l +2 (cid:88) i = 1 λ i ( ε, n ) x i + ε R ( ε, n ) , (6.6)where both Q and R are continuous functions of n ∈ C , including when n ∈ S ◦ E , and rational functions of ε with nonvanishing denominator.Since C ⊂ S d − is compact, they can thus be bounded by an overallconstant independent of ε ∈ (0 , ε ]. This holds even at the singularpoint n ∈ C ∩ S ◦ E when there is cancellation of powers of ε in numeratorand denominator, whence (6.5). Thus, comparing with (6.2), we deducethat, for 0 < ε ≤ ε , the images f ε ( C ) lie in an O( ε ) neighborhood U ε of the edge E . This immediately implies that the limiting set iscontained within the edge: lim ε → + f ε ( C ) ⊂ E . The remaining task isto prove that every point in E is contained in the limit, and thereforelim ε → + f ε ( C ) = E .We already know that both endpoints x , x are contained in thelimiting set. Thus, our remaining task is, given a point x ∈ E ◦ , tofind points y ε ∈ f ε ( C ) that converge to x = lim ε → + y ε . Although it ispossible to do this by a careful analysis of the underlying formulae, weprefer, for later purposes, to use a simple topological proof.To this end, let Z x be the affine hyperplane passing through x thatis orthogonal to E , and define Z x ,ε = Z x ∩ U ε . We claim that thereexists y ε ∈ f ε ( C ) ∩ Z x ,ε . If true, then we have produced the desiredpoints. To prove the claim, observe that U ε \ Z x ,ε consists of two disjointopen subsets, say U x ,ε , U x ,ε with x i ∈ U i x ,ε for i = 1 , 2. Moreover,since we know that all the points in f ε ( C ∩ S ◦ i ) converge to x i , ifwe choose ε sufficiently small, then f ε ( C ) ∩ U i x ,ε (cid:54) = ∅ . Therefore, f ε ( C ) ∩ Z x ,ε = ∅ would contradict the connectedness of f ε ( C ). Thiscontradiction establishes the above claim. We thus conclude that, assets f ε ( C ) −→ E as ε −→ . (6.7)For later purposes, we need slightly more than mere set-theoreticconvergence (6.7). Namely, we require the existence of a continuous set-valued homotopy that connects the images of f ε : C → K for ε > (cid:98) f : C → K with range equal to the edge E = (cid:98) f ( C ), amodel being Example 4.1. Rather than write down an explicit formulafor this homotopy, we will instead construct its graph.Consider the graphΓ = (cid:8) (cid:0) ε, n , f ε ( n ) (cid:1) (cid:12)(cid:12) < ε ≤ ε , n ∈ C (cid:9) ⊂ (0 , ε ] × C × K APS CONVERGING TO CONVEX HULLS 19 of the map F ( ε, n ) = f ε ( n ) for 0 < ε ≤ ε and n ∈ C . Let Γ = Clos Γbe its closure in [0 , ε ] × C × K . According to the preceding proof, Γis the graph of the set-valued map F : [0 , ε ] × C → K given by F ( ε, n ) = f ε ( n ) , ε > , x , ε = 0 , n ∈ C ∩ S ◦ , x , ε = 0 , n ∈ C ∩ S ◦ ,E, ε = 0 , n ∈ C ∩ S ◦ E , (6.8)its final value being the entire edge E ⊂ K . Moreover, since Γ is closedand K is compact Hausdorff, the Closed Graph Theorem for set-valuedfunctions, [1, Prop. 1.4.8], implies that the set-valued function F isupper hemicontinuous, as per Definition 4.2. Thus, for all 0 < ε ≤ ε ,(6.8) defines an upper hemicontinuous homotopy from each f ε : C → K to the set-valued map (cid:98) f : C → K with (cid:98) f ( n ) = F (0 , n ), whose range (cid:98) f ( C ) is the edge E . Remark : An alternative approach, that avoids set-valued homotopiesand, later, set-valued homology, is to “tilt” the subset Γ so that it be-comes a graph by introducing new coordinates on the Cartesian prod-uct space [0 , ε ] × C × K . However, this is more technically tricky toaccomplish in the higher dimensional cases to be handled below, andthe set-theoretic approach provides a cleaner path to the proof.The remainder of the proof works by induction on the dimensionof the face F . Thus, the next case is that of a two-dimensional face F ⊂ B ⊂ R d . The main steps of the proof in this situation willthen be straightforwardly adapted to any higher dimensional face. Let x , . . . , x k be the vertices of F and let E , . . . , E k be its edges. We labelthe vertices and edges so that E j connects x j to x j +1 , with indices takenmodulo k throughout, whence x k +1 = x . Thus E = (cid:83) kj =1 E j = ∂F is the polygonal boundary of F . We assume that there are l ≥ x k +1 , . . . , x k + l ∈ F \{ x , . . . , x k } , while the remainingpoints in the configuration x k + l +1 , . . . , x n ∈ K \ F . Keep in mind that F is convex.Let S i , (cid:101) S j , S F be the normal spherical polytopes of x i , E j , F , respec-tively, so that (cid:101) S j ⊂ ∂S j ∩ ∂S j +1 and S F ⊂ ∂ (cid:101) S j for all j = 1 , . . . , k . Let W F ⊂ S d − be the open set (3.11), (3.12), and let (cid:101) N ⊂ N ⊂ W F be thetwo-dimensional submanifolds satisfying the hypotheses of Proposition k + l (cid:88) i = 1 λ i ( ε, n ) = 1 + ε Q ( ε, n ) , f ε ( n ) = k + l (cid:88) i = 1 λ i ( ε, n ) x i + εR ( ε, n ) , (6.9)where both Q and R are continuous functions of n ∈ N , and rationalfunctions of ε with nonvanishing denominator. These formulae againimply that the images f ε ( N ) lie in an O( ε ) neighborhood U ε of the face F , and hence lim ε → + f ε ( N ) ⊂ F . The remaining task is to prove thatevery point in F is contained in the limit, a result that requires a moresophisticated topological argument than in the curve case.For this purpose, we replace N by (cid:101) N . Clearly, if we can provelim ε → + f ε ( (cid:101) N ) = F , by the preceding result the same is true of N ⊃ (cid:101) N .According to Proposition 3.3, (cid:101) N ∩ S ◦ j (cid:54) = ∅ and (cid:101) N ∩ (cid:101) S ◦ j (cid:54) = ∅ for all j = 1 , . . . , k and either n F ∈ (cid:101) N when d = 3, where n F is the unit out-ward normal to the polyhedral facet F , or (cid:101) N ∩ S ◦ F (cid:54) = ∅ when d > L = ∂ (cid:101) N can be decomposed into nonover-lapping curves L , . . . , L k that satisfy L j ⊂ S ◦ j ∪ (cid:101) S ◦ j ∪ S ◦ j +1 , againmodulo k . Let { n j } = L j − ∩ L j ⊂ S ◦ j denote the common endpointsof adjacent curves in L = ∂ (cid:101) N .Let us set I = [ 0 , ε ] for ε > f ε ( L j ) → E j as sets and, moreover, there existsan upper hemicontinuous homotopy (of set-valued mappings) from each f ε : L j → K for all 0 < ε ≤ ε to the set-valued limit (cid:98) f : L j → K withrange equal to the edge E j = (cid:98) f ( L j ). The graph of this homotopy,Γ j ⊂ I × L j × K ⊂ I × S × K, is a closed subset of the indicated Cartesian product space.We now piece together these homotopy graphs to define a homotopyfrom f ε ( L ) to E = ∂F whose graph isΓ = k (cid:91) j =1 Γ j ⊂ I × L × K ⊂ I × S × K. (6.10)Note that Γ is a closed subset that defines the graph of an upper hemi-continuous function because each Γ j is closed and, moreover,Γ j − ∩ (cid:0) I × { n j } × K (cid:1) = Γ j ∩ (cid:0) I × { n j } × K (cid:1) , including when ε = 0, since f ε ( n j ) → x j , and soΓ j − ∩ (cid:0) { } × { n j } × K (cid:1) = { (0 , n j , x j ) } = Γ j ∩ (cid:0) { } × { n j } × K (cid:1) . APS CONVERGING TO CONVEX HULLS 21 Given x ∈ F ◦ , we seek y ε ∈ f ε ( (cid:101) N ) that converge to x as ε → + . Let Z x be the affine subspace of dimension d − x that isorthogonal to F . Define Z x ,ε = Z x ∩ U ε . Again, if we can prove thereexists y ε ∈ Z x ,ε ∩ f ε ( (cid:101) N ), we are done. Suppose not, i.e., suppose that f ε ( (cid:101) N ) ⊂ U ε \ Z x ,ε . The idea is to demonstrate that this is topologicallyimpossible due to the contractibility of (cid:101) N , and hence contractibilityof f ε ( (cid:101) N ), whereas f ε ( L ) = f ε ( ∂ (cid:101) N ), for ε sufficiently small, defines anontrivial homology class in U ε \ Z x ,ε .If we were dealing with ordinary mappings, this topological argumentwould be straightforward. But because (cid:98) f is a set-valued mapping,we will need some more sophisticated tools from set-valued algebraictopology to establish the contradiction. We summarize the basic the-ory, based on a paper of Yongxin Li, [9], in Appendix A. In accordancewith the notation introduced there, we use roman H n ( X ) to denote thestandard n -th order singular homology groups of a topological space X , and calligraphic H n ( X, U ) to denote the corresponding n -th orderset-valued homology groups relative to a chosen open cover U . (Asnoted in the appendix, if one does not choose this cover carefully, theset-valued homology groups are all trivial, and would hence be uselessfor our purposes.)In our situation, we select the particular open covering V of U ε \ Z x ; ε consisting of all open sets of the form V = H ∩ (cid:0) U ε \ Z x ; ε (cid:1) such that x ∈ ∂V, (6.11)where H is an open half-space in R d . We claim that V satisfies Li’scontractible finite intersection property, because the intersection of anyfinite collection of such open sets, if non-empty, is homeomorphic tothe Cartesian product of an open ( d − (cid:98) f : L → E , whose range is thepolygonal boundary of the face (cid:98) f ( L ) = E = ∂F , is compatible withthe open covering V , because (cid:98) f ( n ) is either a vertex or an edge E j .Moreover, when ε > 0, the map f ε is continuous and single-valued,which implies trivially that its restriction to L is compatible with anyopen covering of U ε \ Z x ; ε .The family of maps { f ε , (cid:98) f } thus defines, by varying ε , an upperhemicontinuous homotopy of multi-valued functions with closed val-ues. It follows from [9, Prop. 6] that the upper triangle in Figure 4commutes. In the same figure, the square is divided into two triangles.It follows from the definitions that the top right triangle in the square H ( ∂ (cid:101) N ) H ( U ε \ Z x ; ε , V ) H ( U ε \ Z x ; ε , V ) H ( U ε \ Z x ; ε ) H ( (cid:101) N ) = ( (cid:98) f ) ∗ ( f ε ) ∗ f ε, ∗ Id i (cid:93) ι ∗ f ε, ∗ Figure 4. Commutative Diagramcommutes. As noted in the Appendix A, the map i (cid:93) on the bottomright is an isomorphism. Finally, the map ι : L = ∂ (cid:101) N → (cid:101) N denotesthe inclusion map, and so it is a standard fact from ordinary singularhomology theory that the bottom left triangle commutes.Let 0 (cid:54) = c = [ ∂ (cid:101) N ] ∈ H ( ∂ (cid:101) N ) be the homology class representing ∂ (cid:101) N ,which is, in fact, a generator. We claim that ( (cid:98) f ) ∗ ( c ) is a non-zeroelement of H ( U ε \ Z x ; ε , V ). Indeed, ( (cid:98) f ) ∗ ( c ) = i (cid:93) (cid:0) [ E ] (cid:1) (cid:54) = 0 since thehomology class [ E ] = [ ∂F ] ∈ H ( U ε \ Z x ; ε ) is nonzero and i (cid:93) is anisomorphism. This thus proves the claim that( f ε ) ∗ ( c ) = ( (cid:98) f ) ∗ ( c ) (cid:54) = 0 , and hence f ε, ∗ ( c ) = i − (cid:93) (cid:2) ( f ε ) ∗ ( c ) (cid:3) (cid:54) = 0 . On the other hand, the bottom left triangle in the square in figure 4shows that f ε, ∗ vanishes identically on H ( ∂ (cid:101) N ), so that f ε, ∗ ( c ) = 0, thusleading to the desired contradiction and thus establishing the existenceof y ε ∈ f ε ( (cid:101) N ). This finishes the proof that every point of F belongs tothe inner limit of f ε ( (cid:101) N ), as ε → 0. We conclude that both f ε ( (cid:101) N ) and f ε ( N ) → F as sets as ε → + .Finally, to establish the existence of a set-valued homotopy connect-ing the maps f ε : N → K to the set-valued map (cid:98) f : N → K with range F = (cid:98) f ( N ), we proceed as follows. As in the curve case, we construct As in Appendix A, the parentheses indicate the induced maps on set-theoretichomology. APS CONVERGING TO CONVEX HULLS 23 its graph Γ ⊂ I × N × K as the closure of the graphΓ = { ( ε, n , f ε ( n ) | n ∈ N, < ε ≤ ε } ⊂ (0 , ε ] × N × K of the continuous map F ( ε, n ) = f ε ( n ). Again, by the Closed GraphTheorem for set-valued functions, Γ is the graph of an upper hemicon-tinuous set-valued function F : I × N → K , which, for 0 < ε ≤ ε ,defines the required upper hemicontinuous homotopy.Finally, let us outline the proof in the general case. To this end, weestablish the following result by induction on the dimension m of theface, using the preceding two-dimensional case as a model. Proposition 6.1. Let F ⊂ B be an m -dimensional face, and let N ⊂ S d − be an m -dimensional submanifold satisfying the conditions ofProposition 3.3. Then, the set-theoretic lim ε → + f ε ( N ) = F . Moreover,for ε > sufficiently small, there is a continuous set valued homotopyfrom f ε : N → K to the set-valued map (cid:98) f : N → K whose range is theentire face: f ( N ) = F . Referring back to the preceding argument for two-dimensional polyg-onal faces, the key formulae (6.9) work exactly as before, with x , . . . , x k the vertices of F and x k +1 , . . . , x k + l additional points in the configu-ration, if any, in F \ { x , . . . , x k } . These in turn imply that, for ε sufficiently small, the images f ε ( N ) lie in a O( ε ) neighborhood of F ,thus proving that lim ε → + f ε ( N ) ⊂ F .To prove that every point in x ∈ F is contained in the limit, wereplace N by the open submanifold (cid:101) N ⊂ N given in Proposition 3.3.Again assume the contrary, that f ε ( (cid:101) N ) ⊂ U ε \ Z x ,ε , where Z x ,ε = Z x ∩ U ε with Z x the affine subspace of dimension d − m passing through x orthogonal to F . According to the inductive hypothesis, its ( m − L i ⊂ ∂ (cid:101) N satisfies lim ε → + f ε ( L i ) = G i , the corresponding ( m − F , through anupper hemicontinuous homotopy from f ε : L i → K to the set-valuedmap (cid:98) f : L i → K with range G i = (cid:98) f ( L i ). We then, as in (6.10),piece together these subface homotopies so as to construct an upperhemicontinuous homotopy from f ε : ∂ (cid:101) N → K to the set-valued map (cid:98) f : ∂ (cid:101) N → K whose range is all of ∂F = (cid:98) f ( ∂ (cid:101) N ).The topological argument then proceeds in an identical manner, theonly difference being that the open cover V is constructed as in (6.11),but now the intersections are homeomorphic to the contractible Carte-sian product of a spherical sector of dimension m with a ball of di-mension d − m . Further, we use the same commutative diagram asin Figure 4 but with the first homology group H replaced by H m − throughout. The resulting topological contradiction proves thatlim ε → + f ε ( (cid:101) N ) = lim ε → + f ε ( N ) = F. Finally, the construction of the corresponding upper hemicontinuousset-valued homotopy from f ε : N → K to (cid:98) f : N → K with range (cid:98) f ( N ) = F proceeds exactly as before.The final step in the proof of the Main Theorem 2.1 is to provethat lim ε → + f ε ( S d − ) = B . For this, we split up B into its facets B = F ∪ · · · ∪ F k . For each F i , by combining Lemma 5.2 withthe argument following (6.9), we deduce that lim ε → + f ε ( S F i ) ⊂ F i ,and hence lim ε → + f ε ( S d − ) ⊂ B . On the other hand, by the case m = d − d − N i ⊂ S d − such that lim ε → + f ε ( N i ) = F i . We conclude thatlim ε → + f ε ( S d − ) = B , as desired. Moreover, we can similarly piecetogether the set-valued homotopies for each facet to find a set-valuedhomotopy from f ε : S d − → K to the inverse Gauss map of the bound-ary, (cid:98) f γ − B : S d − → B . This, at last, completes our proof. Appendix A. Set-Valued Homology In this appendix, we review the basics of set-valued singular homol-ogy following Y. Li, [9]. For simplicity, we will use Q as our ring ofcoefficients throughout.Let X , Y be connected normal Hausdorff topological spaces. Givena set-valued mapping F : X → Y and an open covering U of Y , wesay that F is compatible with U if and only if for any x ∈ X , there issome U ∈ U such that F ( x ) ⊂ U . Define C ( X, Y, U ) = (cid:110) F : X → Y (cid:12)(cid:12)(cid:12) F is an upper hemicontinuous mappingwith closed values compatible with U (cid:27) . (A.1)Let∆ n = (cid:8) x = ( x , . . . , x n ) ∈ R n +1 (cid:12)(cid:12) x i ≥ , x + x + · · · + x n = 1 (cid:9) denote the standard n -dimensional simplex. For i = 0 , . . . , n , let ϕ i ( x , . . . , x n ) = ( x , . . . , x i − , , x i , . . . , x n )map the ( n − n − to the i -th face ∆ ( i ) n =∆ n ∩ { x i = 0 } of the n -dimensional simplex.Given an open cover U of Y , we define the n -th set-valued chain group C n ( Y, U ) to be the free abelian group generated by C (∆ n , Y, U ) and APS CONVERGING TO CONVEX HULLS 25 call it the n -th set-valued chain group. We then define the boundaryoperator ∂ n : C n ( Y, U ) → C n − ( Y, U ) by ∂ n c n = n (cid:88) i =0 ( − n c n ◦ ϕ i . (A.2)Thus, ∂ n ◦ ∂ n +1 = 0, which is usually abbreviated by ∂ = 0.The n -th set-valued homology group of ( Y, U ) is then given by H n ( Y, U ) = Ker ∂ n / Im ∂ n +1 . (A.3)As noted by Li, [9], if one is not careful when choosing the cover U , allset-valued homology groups are trivial, and would thus be of no helpestablishing our desired topological result. To avoid this difficulty, Liimposes the contractible finite intersection property on the cover U .This property requires that the intersection of any finite collection ofelements of the cover is either empty or contractible.Since ordinary functions can be viewed as set-valued functions, thereis a natural inclusion map i from the n -th chain group C n ( Y ), as definedin the usual singular homology theory, to C n ( Y, U ). The inclusion is achain map, and thus induces a group homomorphism i (cid:93) : H n ( Y ) −→ H n ( Y, U ) , (A.4)which, according to [9, Theorem 11], is actually an isomorphism.Moreover, an upper hemicontinuous set-valued mapping F : X → Y with closed values induces a chain map from C n ( X ) to C n ( Y, U ), andthus induces a group homomorphism( F ) ∗ : H n ( X ) −→ H n ( Y, U ) . (A.5)In general, we will place parentheses around ( F ) ∗ in order to distinguishthe set-valued homology group homomorphism from the usual grouphomomorphism f ∗ : H n ( X ) → H n ( Y ) on the corresponding singularhomology groups induced by a continuous (ordinary) function f : X → Y . Further results of Li, [9], are quoted in the text as needed. Acknowledgements. We would like to thank Elias Saleeby, Dennis Sul-livan, Daniele Tampieri, Paolo Emilio Ricci, and Kamal Khuri-Makdisifor their useful remarks and suggestions. References [1] Aubin, J.P., and Frankowska, H., Set-Valued Analysis , Birkh¨auser, Boston,2009.[2] Arutyunov, A.V., and Obukhovskii, V., Convex and Set-Valued Analysis : Se-lected Topics , De Gruyter, Berlin, 2017.[3] Avis, D., Bremner, D., and Seidel, R., How good are convex hull algorithms?, Comput. Geom. 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