Ambrus Gergely

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Theorems from the interface ofconvex geometry and analysisOutline of Ph.D. dissertationGergely AmbrusAdvisor:Dr. Ferenc FodorDoctoral School in Mathematics and Computer ScienceUniversity of SzegedFaculty of Natural Sciences and Information TechnologyBolyai InstituteDepartment of GeometrySzeged2010.1. IntroductionThe dissertation investigates three different problems, which areconnected via the underlying, intuitive geometric motivation. Theresults are obtained by using geometric, combinatorial and analytictools. We note that all the topics discussed here originate from thefirst half of the 20th century, hence they are well embedded in theresearch field of discrete and convex geometry.The dissertation is based on the following three publications.• G. Ambrus, A. Bezdek, F. Fodor, A Helly-type transversal the-orem for n-dimensional unit balls, Archiv der Mathematik 86(2006), no. 5, 470–480.• G. Ambrus, F. Fodor, A new lower bound on the surface areaof a Voronoi polyhedron, Periodica Mathematica Hungarica 53(2006), no. 1-2., 45–58.• G. Ambrus, K. J. Bo¨ro¨czky, Stability results for the volume ofrandom simplices. Submitted for publication, American Journalof Mathematics. pp. 1–26.2. Transversals of unit ballsChapter 1 deals with the following question. Let F be a family ofsets in Rd. We say that a line ` is a transversal to F , if it intersectsevery member of F . If F has a transversal, then it is said to haveproperty T . If every k or fewer members of F have a transversal, thenF has property T (k).1The question is the following: how can we guarantee that propertyT holds? In particular, we would like to derive the validity of T fromT (k) with some k. Such a setting is familiar from Helly’s classicaltheorem, which states that if every at most d+ 1 members of a finitefamily of convex sets in Rd has a common point, then all the sets in thefamily intersect in a common point. Thus, such a transversal theoremcan be understood as a generalisation of Helly’s theorem.It turns out that the above goal is too optimistic, if one considersall families of convex bodies: there exists no such general result. Evenfor families that consist of pairwise disjoint translates of an arbitraryconvex body in R3, no such result exists, as was shown by Holmsenand Matousˇek [HM04].Our work considers the case when F consists of unit balls in Rd.We are typically interested in large d’s. The first related result byHadwiger [Had56] states that for any family of thinly distributed ballsin Rd, the property T (d2) implies T , where a family of balls is thinlydistributed if the distance between the centers of any two balls is atleast twice the the sum of their radii. Prior to our result, in [HKL03]and [CGH05] it was proved that or any family of pairwise disjoint unitballs in R3, T (11) implies T .We impose a condition on the pairwise distances of the centres,which is weaker than Hadwiger’s condition, but stronger than dis-jointness. This will be referred as the distance condition.Theorem 1. Let d > 2, and F be a family of unit balls in Rdwith the property that the mutual distances of the centres are at least2q2 +√2. If every at most d2 members of F have a common linetransversal, then all members do.The methods used to prove Theorem 1 have been pushed furthersince the publication of [ABF06]. After a series of results, Cheong,2Goaoc, Holmsen and Petitjean [CGHP08] proved that for any systemof disjoint unit balls in Rd, T (4d− 1) implies T .The proof of Theorem 1 is based on the following statement. LetB1, . . . , Bm be disjoint unit balls in Rd. Consider the set of all directedlines intersecting B1, . . . , Bm in this order, and denote the set of unitdirection vectors of these lines by K(B1, . . . , Bm).Theorem 2. Let Fd = {B1, . . . , Bm} be a family of unit balls satisfy-ing the distance condition. Then K(B1, . . . , Bm) is strictly sphericallyconvex.The crucial advantage of Theorem 2 is that it reduces the originalproblem to a 3-dimensional one, which can be attacked by standardanalytical tools.After establishing the convexity of the cone of transversal direc-tions, in Section 1.3 we prove that if a family Fd of unit balls satisfyingthe distance condition has a transversal, then all the transversals of Fdintersect the unit balls in the same order (or its reverse). This orderingis called a geometric permutation of Fd. Thus, the distance conditionimplies that there is at most one geometric permutation of Fd.Finally, in Section 1.4, we prove Theorem 1 by using the previousresults and invoking the strong version of the Spherical Helly Theorem.3. A new lower bound for the Strong Dodeca-hedral ConjectureThe contents of Chapter 2 are to give an improvement on the lowerbound on the surface area of a Voronoi cell in a unit ball packing.A family B of unit balls in R3 forms a packing if no two membersof B have a common interior point. We are mostly interested in howdense a packing of unit balls may be, where the density of a packing3is the proportion of the space covered by the balls. We define thisas the limit of the proportion of the volume of the covered part ofa ball, where the centre of the ball is fixed and its radius tends toinfinity. According Kepler’s Conjecture [Kep66], the packing densityof unit balls in R3 is pi/√18 ≈ 0.74078 . . . , which is attained by alattice packing. This result was proved recently by Hales [Hal05].In a ball packing, the Voronoi cell of a ball B ∈ B is the set ofpoints x ∈ R3 with the property that x is closer to the centre of Bthan to any other centre in B. It is well known that Voronoi cells areconvex polyhedra, and we may in fact assume that they are polytopes.The Dodecahedral Conjecture, formulated by L. Fejes To´th [FT43]in 1943, states that the minimal volume of a Voronoi cell in a 3-dimensional unit ball packing is at least as large as the volume of aregular dodecahedron of inradius 1. This problem has been recentlysettled in the affirmative by Hales and McLaughlin [HM]. K. Bezdek[Bez00] phrased the following generalised version.Conjecture (Strong Dodecahedral Conjecture). The minimumsurface area of a Voronoi cell in a unit ball packing in R3 is at leastas large as the surface area of the regular dodecahedron circumscribedabout the unit ball, that is 16.6508 . . ..In Chapter 2, we prove the following statement [AF06].Theorem 3. The surface area of a Voronoi cell in a unit ball packingin R3 is at least 16.1977 . . ..This is currently the best estimate related to the problem. Priorto our result, the strongest bound was given by K. Bezdek and E.Daro´czy-Kiss [BDK05], who, based on Muder’s ideas ([Mud88] and[Mud93]), established the lower bound 16.1445 . . . . Our improvementfollows these lines as well.4In the proof, the cones suspended by the faces of the Voronoi cell arereplaced with cones of special types in such a way, that the surfaceto solid angle ratio does not increase. The obtained configurationsbelong to a restricted class, in which the minimiser of the surface areais found by standard analytic methods.In Section 2.2, the replacement steps are established. The conesused for replacements are the following. A right circular cone (RCC)is a cone whose base is a circular disk and its apex lies on the lineperpendicular to the disk passing through its center. A shaved circleis the intersection of a disk and a convex polygon that contains thecenter of the disk. A shaved right circular cone (SRCC) is a cone whosebase is a shaved circle and its apex lies on the line perpendicular to thedisk and passing through its center. The desired replacements withRCC’s or SRCC’s are achieved via a series of basic replacement steps.Then, in Section 2.3, the surface to solid angle ratio of these specialcones are further approximated.Finally, in Section 2.4, the optimal configuration is determined us-ing the previous approximations by a quite strenuous calculation. Theminimal configuration has 13 identical faces and one face of a smallersolid angle. However, these faces cannot be joined to form a polytope,which accounts for the error between our estimate and the conjecturedextremal value.4. Stability results for the volume of randomsimplicesThe following question serves as the motivation for Chapter 3.Given a convex body K in Rd, what is the expected value of thevolume of a random simplex in K? We work with two (or, rather,three) models: in the first, all the vertices of the simplex are chosen5uniformly and independently from K, while in the second, one vertexis at a fixed position – in a special case, this is γ(K), the centroid ofK. We are interested in other moments as well, and also, we wouldlike the answer to be invariant under affine transformations.Definition. Let K be a convex body in Rd. For any n > d + 1 andp > 0, letEpn(K) = V (K)−n−pZK. . .ZKV ([x1, . . . , xn])p dx1 . . . dxn.Further, for a fixed x ∈ Rd, letEpx(K) = V (K)−d−pZK. . .ZKV ([x, x1, . . . , xd])p dx1 . . . dxd.Specifically, we write Ep∗(K) for Epx(K), when x = γ(K).These quantities have many connections to other concepts; for ex-ample, Sylvester’s problem, the volume of centroid bodies and intersec-tion bodies, the volume of Legendre’s ellipsoid, Busemann’s randomsimplex inequality, the Busemann-Petty centroid inequality, and soon. These links are elucidated in Section 3.1.One is mostly interested in the the minimisers and maximisers ofthe above expectations among convex bodies. The search of thesedates back to the early 20th century, see Blaschke ([Bla17] and [Bla23]).The minimisers are known in full generality.Theorem 4. (Blaschke [Bla23], Busemann [Bus53], Groemer[Gro74])For any convex body K in Rd, for any p > 1, and for any n > d + 1,we haveEpo(K) > Epo(Bd) and Ep∗(K) > Ep∗(Bd) and Epn(K) > Epn(Bd).Here Epo(K) = Epo(Bd) if and only if K is an o-symmetric ellipsoid,6and Ep∗(K) = Ep∗(Bd) or Epn(K) = Epn(Bd) if and only if K is anellipsoid.As for the maximisers, the Simplex conjecture states that for anyconvex body K in Rd, and for any p > 1 and n > d + 1, Ep∗(K) 6Ep∗(T d) and Epn(K) 6 Epn(T d), with equality if and only if K is asimplex. This is verified only in the plane.Theorem 5. ([Bla17],[DL91],[Gia92],[CCG99]) If K ⊂ R2 is aconvex disc, then for any n > 3 and p > 1, Epn(K) 6 Epn(T 2) andEp∗(K) 6 Ep∗(T 2), with equality if and only if K is a triangle.The importance of the Simplex conjecture stems from the fact thatthe affirmative answer to it would imply the Slicing conjecture.In Chapter 3 of the dissertation, based on [AB], we provide thecorresponding stability estimates for Theorems 4 and 5. The resultsare formulated with the use of the Banach-Mazur distance δBM(K,M)of the convex bodies K and M , which is defined by δBM(K,M) =min{λ > 1 : K − x ⊂ Φ(M − y) ⊂ λ(K − x)}, where Φ ∈ GLd andx, y ∈ Rd. Our results are as follows.Theorem 6. If K is a convex body in Rd with δBM(K,Bd) = 1 + δfor δ > 0, then for any p > 1,Ep∗(K) > (1 + γpδd+3)Epo(Bd)Epd+1(K) > (1 + γpδd+3)Epd+1(Bd),where the constant γ > 0 depends on d only. Moreover, if K is cen-trally symmetric, then the error terms can be replaced by γpδ(d+3)/2.7Theorem 7. If K is a planar convex body with δBM(K,T2) = 1 + δfor some δ > 0, and p > 1, thenEp∗(K) 6 (1− cpδ2)Ep∗(T 2)Ep3(K) 6 (1− cpδ2)Ep3(T 2),where c is a positive absolute constant. This estimate is asymptoticallysharp as δ tends to zero.For the proof of Theorem 6, we first assume that K is a symmetricconvex body in John’s position, i.e. the unique ellipsoid of maximalvolume inscribed in K is the unit ball. The core lemma estimatesthe change of the expectation when applying one step of Steiner sym-metrisation in a suitable changed direction. The general result is thenobtained by invoking a recent result of Bo¨ro¨czky [Bo¨r], which esti-mates the Banach-Mazur distance between a convex body K and asymmetric convex body which is obtained by the limit of Steiner sym-metrisations from K. We note that the bound of Theorem 6 is almostasymptotically sharp in terms of δ: there is an example, where theerror is of order ε(d+1)/2.The stability version of the maximum inequality, Theorem 7 in theplane is obtained by the method of linear shadow systems, that wereintroduced by Campi, Colesanti and Gronchi [CCG99]. We assumethat the triangle inscribed in K of maximal area is an equilateral tri-angle. With the aid of basic linear shadow systems, first we reduce theproblem to polygons with at most 6 vertices. The main difficulty liesin the further modification of these polygons. The desired inequalityis then obtained by a technical argument.To conclude the chapter, in Section 3.6 we derive the stability ver-sion of the Petty projection inequality from Theorem 6.8References[AB] G. Ambrus and K. J. Bo¨ro¨czky. Stability results for the volume of randomsimplices. Submitted to the American Journal of Mathematics.[ABF06] G. Ambrus, A. Bezdek, and F. Fodor. A Helly-type transversal theorem forn-dimensional unit balls. Arch. Math. (Basel), 86(5):470–480, 2006.[AF06] G. Ambrus and F. Fodor. A new lower bound on the surface area of a Voronoipolyhedron. Period. Math. Hungar., 53(1-2):45–58, 2006.[BDK05] Ka´roly Bezdek and Endre Daro´czy-Kiss. Finding the best face on a Voronoipolyhedron—the strong dodecahedral conjecture revisited. Monatsh. Math.,145(3):191–206, 2005.[Bez00] Ka´roly Bezdek. On a stronger form of Rogers’s lemma and the minimumsurface area of Voronoi cells in unit ball packings. J. Reine Angew. Math.,518:131–143, 2000.[Bla17] W. Blaschke. U¨ber affine Geometrie IX: Verschiedene Bemerkungen und Auf-gaben. Ber. Verh. Sa¨chs. Akad. Wiss. Leipzig Math.–Phys. Kl., 69:412–420,1917.[Bla23] W. Blaschke. Vorlesungen u¨ber Differentialgeometrie II: Affine Differentialge-ometrie. Springer, Berlin, 1923.[Bo¨r] K.J. Bo¨ro¨czky. Stability of the Blaschke-Santalo´ and the affine isoperimetricinequality. Adv. Math. Accepted.[Bus53] Herbert Busemann. Volume in terms of concurrent cross-sections. Pacific J.Math., 3:1–12, 1953.[CCG99] Stefano Campi, Andrea Colesanti, and Paolo Gronchi. A note on Sylvester’sproblem for random polytopes in a convex body. Rend. Istit. Mat. Univ.Trieste, 31(1-2):79–94, 1999.9[CGH05] Otfried Cheong, Xavier Goaoc, and Andreas Holmsen. Hadwiger and Helly-type theorems for disjoint unit spheres in R3. In Computational Geometry(SCG’05), pages 10–15. ACM, New York, 2005.[CGHP08] Otfried Cheong, Xavier Goaoc, Andreas Holmsen, and Sylvain Petitjean.Helly-type theorems for line transversals to disjoint unit balls. Discrete Com-put. Geom., 39(1-3):194–212, 2008.[DL91] L. Dalla and D. G. Larman. Volumes of a random polytope in a convexset. In Applied geometry and discrete mathematics, volume 4 of DIMACSSer. Discrete Math. Theoret. Comput. Sci., pages 175–180. Amer. Math. Soc.,Providence, RI, 1991.[FT43] L. Fejes To´th. U¨ber die dichteste Kugellagerung. Math. Z., 48:676–684, 1943.[Gia92] Apostolos A. Giannopoulos. On the mean value of the area of a randompolygon in a plane convex body. Mathematika, 39(2):279–290, 1992.[Gro74] H. Groemer. On the mean value of the volume of a random polytope in aconvex set. Arch. Math. (Basel), 25:86–90, 1974.[Had56] H. Hadwiger. Problem 107. Nieuw Arch.Wiskunde, 3(4):57, 1956. Solution:Wiskundige Opgaven 20 (1957), 27–29.[Hal05] Thomas C. Hales. A proof of the Kepler conjecture. Ann. of Math. (2),162(3):1065–1185, 2005.[HKL03] Andreas Holmsen, Meir Katchalski, and Ted Lewis. A Helly-type theorem forline transversals to disjoint unit balls. Discrete Comput. Geom., 29(4):595–602,2003.[HM] T. C. Hales and S. McLaughlin. A proof of the dodecahedral conjecture.http://arXive:math.MG/9811079.[HM04] Andreas Holmsen and Jiˇr´ı Matousˇek. No Helly theorem for stabbing translatesby lines in R3. Discrete Comput. Geom., 31(3):405–410, 2004.[Kep66] J. Kepler. Strena seu de nive sexangula. Tampach, Frankfurt, 1611. In English:The Six-Cornered Snowflake, Oxford, 1966.[Mud88] Douglas J. Muder. Putting the best face on a Vorono˘ı polyhedron. Proc.London Math. Soc. (3), 56(2):329–348, 1988.[Mud93] Douglas J. Muder. A new bound on the local density of sphere packings.Discrete Comput. Geom., 10(4):351–375, 1993.10

Tételek a konvex geometria és az analízis határán

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Tételek a konvex geometria és az analízis határán

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