Geometry is a classical core part of mathematics which, with its birth, marked the beginning of the mathematical sciences. Thus, not surprisingly, geometry has played a key role in many important developments of mathematics in the past, as well as in present times. While focusing on modern mathematics, one has to emphasize the increasing role of discrete mathematics, or equivalently, the broad movement to establish discrete analogues of major components of mathematics.
In this way, the works of a number of outstanding mathema- cians including H. Ladda ned. Spara som favorit. Skickas inom vardagar. This monograph gives a short introduction to the relevant modern parts of discrete geometry, in addition to leading the reader to the frontiers of geometric research on sphere arrangements. The readership is aimed at advanced undergraduate and early graduate students, as well as interested researchers.
Thus, the claim follows from 1 of Lemma 7. Proof: Notice that 1 of Lemma 7. The following statement is rather natural from the point of view of the local geometry introduced above, however, its three-page proof based on Lemma 7. As before, let S denote the boundary of B. First, we dissect P into d-dimensional Rogers simplices. Second,qwe group the d-dimensional Rogers simplices of P as follows. Namely, Lemma 7. Finally, b 3 follows with the help of 2 of Lemma 7. We regard all points in Ed as row vectors and use qT for the column vector that is the transpose of the row vector q.
Moreover, [q1 ,. The following is clear. Based on Definition 7. The following are some simple properties of the volume force. We leave the rather straightforward proofs to the reader. Then the following hold. For more details and examples on volume forces we refer the interested reader to the proper sections in . Let G be a graph defined on this vertex set p. Here, G may or may not consist of the edges of P. We think of the edges of G as defining those pairs of vertices of P that are constrained not to get closer.
In the terminology of the geometry of rigid tensegrity frameworks each edge of G is a strut. For more information on rigid tensegrity frameworks and the basic terminology used there we refer the interested reader to . Let e denote the number of edges of G. Another way to introduce R p is the following. Now, we can rewrite xj k2 ,. Then it is immediate that 21 dx the inequalities of 7. Proof: The assumptions, 7. See  for more information. Here bars mean edges whose lengths are constrained not to change. We say that G p resp. Proof: By Theorem 7. However, then by the infinitesimal rigidity of G p , this would imply that p0 is trivial.
The following definition leads us to the core part of this section. Then P is strictly locally volume expanding over G. Proof: The inequalities 7. Such k exists by the argument in  as well as . Because 7. All but a finite number of tiles are fixed. Each Pi is strictly locally volume expanding, therefore the volume of each of the tiles must be fixed. But the strict condition implies that the motion of each tile must be an isometry.
Because the tiling is face-to-face and the vertices are given by GP we conclude inductively on the number of tiles that each vertex of GP must be fixed. Proof: Without loss of generality R y we may assume that f is differentiable. This completes our proof of Lemma 8. Also, it follows from the given setup in a straightforward way that! Moreover, f is positive and monotone increasing on 0, 1], and by the Brunn— 1 Minkowski inequality see, e.
Thus, Lemma 8. This finishes the proof of Theorem 2. We do not prove it here, but instead refer the interested reader to the numerous resources on that, in particular to , , , and . Theorem 9. Moreover, for sufficiently large d, 5d can be replaced by 4d. Now, the proof of Corollary 3.
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Lemma 9. The latter inequality together with Lemma 9. Based on Lemma 9. By assumption the radius of BF is at most r. This completes the proof of the first part of Theorem 3. Finally, the second part of Theorem 3. This is a somewhat stronger statement than the first claim of Theorem 3. Now, let us take the following special class of convex sets in E3. Then B[Y ] can be X-rayed by 3 pairwise orthogonal directions of E3 , one of which can be chosen arbitrarily. We need the following lemma. Proof: We may assume that Y is closed.
This finishes the proof of Lemma 9. We call L1 vertical, and lines perpendicular to L1 horizontal. We pick two pairwise orthogonal, horizontal lines, L2 and L3. By Lemma 9. Furthermore, each vertex is necessarily the midpoint of the quarter arc of the great circle on which it lies, and NB[Y ] b does not intersect either of the three great circles in any other point.
We call these tiles blocking tiles. Now, by rotating L2 and L3 together in the horizontal plane, we can easily avoid all the blocking tiles; that is, we can find a rotation R about the line L1 such that none of the blocking tiles has a vertex on both circles C R L2 and C R L3. Based on 3 Theorem 3. In order to achieve this let us take two regular hexagons of edge length 1 inscribed into S3 such that their 2-dimensional planes are totally orthogonal to each other in E4.
Now, let P be the convex hull of the 12 vertices of the two regular hexagons. If F is any facet of P, then it is easy to see that F is a 3-dimensional simplex having two pairs of vertices belonging to different hexagons with the property 9. Let us take a 2-dimensional plane E2 and a 3-dimensional subspace E3 in E5 such that they are totally orthogonal to each other with both passing through the origin o of E5. For the details of the construction of P3 see . If F is any facet of P, then it is easy to see that F is a 4-dimensional simplex having two vertices in P2 and three vertices in P3.
If one projects F from the center o of S4 onto S4 , then the projection F 0 is a 4-dimensional spherical symplex. Among its five vertices there are two vertices say, a and b lying in P2. Now, let c0 be the center of C and c be an arbitrary point of C; moreover, let m be the midpoint of spherical segment ab. If s denotes the center of the circumscribed sphere of F 0 in S4 , then s is a point of the spherical segment c0 m. In order to achieve this let us take two 3-dimensional subspaces E31 and E32 in E6 such that they are totally orthogonal to each other with both passing through the origin o of E6.
If one projects F from the center o of S5 onto S5 , then the projection F 0 is a 5-dimensional spherical symplex. It follows from the construction above that two spherical triangles formed by the two proper triplets of the vertices of F 0 have circumscribed circles C1 and C2 of radius Rc. The canonical basis of Rd we denote by e1 ,. Given points x1 ,. Given a convex body K in Rd we denote its volume by vold K. Let K be an o-symmetric convex body in Rd.
This completes the proof of the planar case. Now, we handle the 3-dimensional case. In fact, the proof given for Case c also works for Case b , however, Case b is much simpler, so we have decided to consider it separately. For the next two cases we need the following notation. By bi we denote the spherical area of Ci. Let ni denote the number of sides of Fi and let ai stand for the spherical area of Fi. Thus, without loss of generality we can assume that Bd2 is the ellipsoid of minimal volume for K.
The upper bound mentioned there has been derived earlier as a rather simple observation. Corollary 9. In  and , M.
Rudelson proved the following theorem see Corollary 4. It is a standard observation cf. Proof: Let P be a polytope given by Corollary 9. Thus, by Lemma 9. Proof: Let H1 resp. If H1 and H2 are translates of each other, then the claim is obviously true. Let E2 be the 2-dimensional linear subspace of Ed that is orthogonal to L. In other words, it is sufficient to prove Lemma We prove Theorem 4. In order to simplify matters let us assume that the origin o of Ed belongs to the interior of C.
Hence, applying Lemma Hence, As the case, when the optimal partition is achieved, follows directly from the definition of the nth successive C-inradius of K, the proof of Theorem 4. Theorem Let K be a convex body in Ed and E be a k-dimensional linear subspace of Ed. Thus, applying Theorem Moreover, let S x denote the sphere of radius x centered at o.
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The following statement can be obtained by combining Corollary 3. Lemma This implies, via a classical theorem of convexity see, e. The sets Wij and W ij are the walls between the nearest point and farthest point Voronoi cells. Clearly, Then the volume functions Vd t, s and V d t, s are continuously differentiable in t and s simultaneously, and for any fixed t, the nearest point and farthest point Voronoi cells are constant.
Then recall that the point x belongs to the Voronoi cell Vi t0 , s resp. So each Vi t0 , s and Vi t0 , s is a constant function of s. As p t is continuously differentiable, therefore the partial derivatives of Vd t, s and V d t, s with respect to t exist and are continuous by Theorem 5. Thus, Vd t, s and V d t, s are both continuously differentiable with respect to t and s simultaneously. Moreover, clearly there are only a finite number of values of t in the interval [0, 1], where the combinatorial type of the above Voronoi cells changes.
The volume of the truncated Voronoi cells Vi ri s and Vi ri s are obtained from the volume of the d-dimensional Euclidean ball of radius ri s by removing or adding the volumes of the regions obtained by conning over the walls Wij pi t , ri s or W ij pi t , ri s from the point pi t. So, for any fixed t the ball of radius ri s will not be tangent to any of the faces Vi or Vi for all but a finite number of values of s. Thus, we define U to be the set of those t, s , where for some open interval about t in [0, 1], the combinatorial type of the Voronoi cells is constant and for all i, the ball of radius ri s is not tangent to any of the faces of Vi or Vi.
Thus, the formulas for the mixed partial derivatives in Lemma Note also that here we could interchange the order of partial differentiation with respect to the variables t and s. We show a contradiction. If we perturb s slightly to s0 say, then using the formulas for the mixed partial derivatives in Lemma This completes the proof of Lemma Now, recall the construction of the following truncated Voronoi cells. We call the set Vi ri , d resp. Thus, by Lemma A similar calculation works for vold V ri , d. Corollary For more information on the background of this theorem we refer the interested reader to .
Next, we recall Theorem We need to apply this to a sphere, rather than Euclidean space. So we need the following. Proof: Apply Theorem Then the following holds. The result then follows from Lemma Let X be any integrable set in Sn. Then By Corollary Recall that X d p and X d q are the intersections of the closed hemispheres centered at the points of p and q in Sd. Hence, by Lemma This finishes the proof of Theorem 5.
We need the following property for our proof of Theorem 5. It is useful to introduce the notations RN Here we assume that v0 and v then the existence of v01 is trivial. Proof: Lemma This finishes the proof of Lemma Now, we are ready to give a proof of the following volume monotonicity property of hyperbolic simplices. If each inner dihedral angle of Q is at least as large as the corresponding inner dihedral angle of P, then the n-dimensional hyperbolic volume of P is at least as large as that of Q.
Proof: By moving to the space of Gram matrices of n-dimensional hyperbolic simplices and then applying Lemma First, recall the following classical theorem of Andreev . Second, observe that the Andreev theorem implies that the space of the inner dihedral angles of nonobtuse-angled compact convex polyhedra of a given simple combinatorial type different from that of a tetrahedron in H3 is a convex set. As a result we get that if P and Q are given as in Theorem 5. This is a generalization of the analogue 2-dimensional statement proved in .
Then F cannot be translated into the interior of C if and only if the following two conditions hold. We are left with the case when C is not necessarily a smooth convex body in Ed. This implies that by connecting the consecutive points of pi1 , pi2 ,. According to Lemma By the assumption, we have r0 12 Selected Proofs on Ball-Polyhedra Recall the following strong version of spherical convexity. We denote it by Sconv X, S k c, r. Since X is closed, its distance from the closed hemisphere containing y is positive.
As Bd [q0 , 1] separates X from y, the latter is not in convs X , a contradiction. Recall that the intersection of the d-dimensional closed unit balls of Ed centered at the points of X is denoted by B[X]. Thus, i and ii follow. By compactness, there is a sequence i 0 12 Selected Proofs on Ball-Polyhedra For the following investigations it is more proper to use the normal images than the Gauss images of the boundary points of B[X] defined as follows.
Here we refer to the strong version of spherical convexity introduced for Lemma We need to introduce the following notation as follows. Because the proof of this claim is straightforward we leave it to the reader. For more insight on illumination we refer the interested reader to  and the relevant references listed there.
Finally, we need to recall some further notations as well. Finally, let X be a set T in Ed. Now, we are ready to state Lemma Then ii of Lemma He has proved in  that any set of diameter 1 is contained in a complete set of diameter 1. Moreover, he has shown in  that a compact set of diameter 1 in Ed is complete if and only if it is of constant width 1. These facts together with the easy observation that any convex body of constant width 1 in Ed is in fact a spindle convex set, imply that X is contained in a convex body of convex width 1 and any such convex body must necessarily contain convs X.
By Lemma Let the directions u1 , u2 ,. Thus, the probability that We claim that the directions v1 , v2 ,. We show that at least one of the directions v1 , v2 ,. As the spherical caps A1 , A2 ,. As this improves the estimate of Theorem 6. Then recall that the Gauss image of b with respect to K is the set of outward unit normal vectors of hyperplanes that support K at b. Then the faces of P form the closed cells of a finite CW-decomposition of the boundary of P. The relative interior resp. Clearly, S b is a support sphere of P. This also shows that b belongs to the relative interior of F.
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Hence, the union of the relative interiors of the faces covers bdP. It is sufficient to show that F is at most m-dimensional. Hence, b belongs to the relative interior of a face of smaller dimension. This concludes the proof of Theorem Proof: because the faces form a CW-decomposition of the boundary of P, there is a vertex v. So, v belongs to at least d generating spheres from the family of generating balls. We denote the centers of those spheres by x1 , x2 ,. Proof: First, we show that the intersection of two faces F1 and F2 is another face or the empty set.
The intersection of the two supporting spheres that intersect P in F1 and F2 is another supporting sphere of P, say S l p, r. From this the existence of a unique maximum common lower bound i. Using Theorem Clearly, P has facets. Then r and s belong to the relative boundary of F. Hence, by Theorem The proof of the second statement goes as follows. Proof: It follows from Theorem Agarwal and J. Aigner and G.
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Banaszczyk, A. Litvak, A. Pajor, and S. Szarek, The flatness theorem for nonsymmetric convex bodies via the local theory of Banach spaces, Math. Bang, On covering by parallel-strips, Mat. Bannai and N. Sloane, Uniqueness of certain spherical codes, Canad. Baranovskii, On packing n-dimensional Euclidean space by equal spheres, Iz.
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Kiss, and M. Mollard, An illumination problem for zonoids, Israel J. Bezdek and T. Hausel, On the number of lattice hyperplanes which are needed to cover the lattice points of a convex body, in Intuitive Geometry, Coll. Bolyai 63, North-Holland, Amsterdam, , 27— Zamfirescu, A characterization of 3-dimensional convex sets with an infinite X-ray number, in Intuitive Geometry, Coll. Bolyai 63, North-Holland, Amsterdam, , 33— Dedicata 68 , 29— Bezdek, E. Liu, Voronoi polyhedra of unit ball packings with small surface area, Periodica Math.
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Bezdek and Gy. Kiss, On the X-ray number of almost smooth convex bodies and of convex bodies of constant width, Canadian Math. MG] March 26, , 1— Bezdek, On a generalization of the Blaschke—Lebesgue theorem for diskpolygons, arXiv MG] March 31, , 1— Boltyanski, The problem of illuminating the boundary of a convex body, Izv. Boltyanskii and M. Boltyanski, A solution of the illumination problem for belt bodies, Mat.
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