Covering functionals of cones and double cones

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1 Wu and Xu Journal of Inequalities and Applications 08) 08:86 RESEARCH Open Access Covering functionals of cones and double cones Senlin Wu* and Ke Xu * Correspondence: wusenlin@nuc.edu.cn Department of Applied Mathematics, Harbin University of Science and Technology, Harbin, China Abstract The least positive number γ such that a convex body K can be covered by m translates of γ K is called the covering functional of K with respect to m), and it is denoted by m K). Estimating covering functionals of convex bodies is an important part of Chuanming Zong s quantitative program for attacking Hadwiger s covering conjecture. Estimations of covering functionals of cones and double cones, which are best possible for certain pairs of m and K, are presented. MSC: Primary 5A0; secondary 5A5; 5A40; 5C7 Keywords: Banach Mazur distance; Compact convex cones; Convex body; Covering; Hadwiger s covering conjecture Introduction A compact convex set K Rn having interior points is called a convex body. The interior and boundary of K are denoted by int K and bd K, respectively. We write Kn for the set of convex bodies in Rn. Concerning the least number ck) of translates of int K needed to cover a convex body K, there is a long-standing conjecture: Conjecture Hadwiger s covering conjecture) For each K Kn, ck) is bounded from above by n, and this upper bound is attained only by parallelotopes. We refer to [ 4], and [5] for more information and references about this conjecture. Note that, for each K Kn, ck) equals the least number of smaller homothetic copies of K needed to cover K see, e.g., [, p. 6, Theorem 4.]). Therefore, ck) m for some m Z+ if and only if m K) <, where m K) is defined by m K) = min γ > 0 : {xi : i =,..., m} R s.t. K n m xi + γ K) i= and called the covering functional of K with respect to m. A closely related concept is studied in functional analysis. Given a bounded subset M of a normed space E with unit ball B, the m-th entropy number εm M) of M is defined by cf. [6, p. 6 7]) m εm M) = inf ε > 0 : M εb + xi ) holds for suitable x,..., xm E. i= The Authors) 08. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Wu and Xu Journal of Inequalities and Applications 08) 08:86 Page of 9 When K is a translate of the unit ball B of a finite-dimensional normed space E, wehave Ɣ m K)=ε m K). Clearly, for each m Z +, Ɣ m ) is affinely invariant. More precisely, Ɣ m K)=Ɣ m TK) ), T A n, where A n is the set of non-degenerate affine transformations from R n to R n.thuswe identify convex bodies that are affinely equivalent, and when writing K n we are actually referring to the quotient space of K n with respect to affine equivalence. For each pair K, K of convex bodies in K n,thebanach Mazur distance d BM K, K ) alsocalledtheasplund metric, cf.[7]) between them is defined by d BM K, K ):=ln min { γ :K TK ) γ K + x, x R n, T A n}. Then K n, d BM ) is a compact metric space cf. [8]and[7]). Zong cf. [9]) proved that Ɣ m ) is uniformly continuous on K n. Bezdek and Khan improved this result by showing that Ɣ m ) is Lipschitz continuous on K n with n )/ ln n) as a Lipschitz constant cf. [0]). These results show that each K K n can be covered by at most n smaller homothetic copies of K if and only if cn):=sup { Ɣ nk):k K n} <. Due to these facts, estimating covering functionals of convex bodies is an important part of Zong s quantitative program for attacking Hadwiger s covering conjecture cf. [9] for more details). Our starting point is Theorem in [9]: Ɣ 8 C) when C is a three-dimensional convex cone the convex hull of the union of a planar convex body K and a singleton not contained in the plane containing K). By applying new ideas, we show that this estimation can be improved when Ɣ 7 K)<, and that this estimation can be extended to higher dimensional situations. Moreover, we also obtain an estimation of Ɣ m C)whenC is a double cone. Covering functionals of convex cones We start with an elementary lemma. Lemma Let K R n be a convex set, x R n, and λ, γ [0, ]. Then x + γ K) K λx +λγ + λ)k. Proof Let z be an arbitrary point in x + γ K) K.Thenz x γ K and z K. Therefore z λx = λz x)+ λ)z λγ K + λ)k =λγ + λ)k. It follows that z λx +λγ + λ)k.

3 Wu and Xu Journal of Inequalities and Applications 08) 08:86 Page of 9 For each x R n,weputx = x,0) R n+. Each point x R n+ canbewrittenintheform x, α), where x R n and α R. LetK beaconvexbodyink n containing the origin o in its interior, and p be a point in R n+ \ R n {0}.PutC = convk {0}) {p}), i.e., C is the cone having p as a vertex and whose base is K. Lemma Suppose that m Z +, γ, λ 0, ), μ [λ,], z = μȳ,0) + μ)p forsome ȳ K, and that {ū i : i [m]} R n is a set of points satisfying K ū i + γ K). Thenthere exists such that z λū i,0)+λγ + λ)c. Proof We have z = μȳ,0)+ μ)p = μȳ,0)+ μ λ)p λ = μ ) μ λ λȳ,0)+ λ)p + ȳ, 0). ) λ λ Without loss of generality we may assume that ȳ ū + γ K. It follows that λȳ,0)+ λ)p λū,0)+λγ K {0} ) + λ)p ) λγ ) λ = λū,0)+λγ + λ) K {0} + λγ + λ λγ + λ p λū,0)+λγ + λ)c. ) By Lemma,wehave ȳ ū + γ K) K λū +λγ + λ)k, which implies that ȳ,0) λū,0)+λγ + λ) K {0} ) λū,0)+λγ + λ)c. ) From ), ), and ) it follows that z λū,0)+λγ + λ)c. For two numbers satisfying 0 λ λ, we put C λ,λ = { λ x,0)+ λ)p : x K, λ [λ, λ ] }. It is not difficult to verify that C = C 0,. And when 0 λ λ andλ 0,wehave C λ,λ = { λ x,0)+ λ)p : x K, λ [λ, λ ] } { λ = λ x,0)+ λ } p : x K, λ [λ, λ ] λ λ

4 Wu and Xu Journal of Inequalities and Applications 08) 08:86 Page 4 of 9 { ) λ = λ x,0)+ λλ p + λ ) } p : x K, λ [λ, λ ] λ } + λ )p { ) λ = λ x,0)+ λλ p : x K, λ [λ, λ ] λ [ ]} λ = λ {λ x,0)+ λ)p : x K, λ, + λ )p λ = λ C λ /λ, + λ )p. 4) Lemma Let m be a positive integer satisfying γ = Ɣ m K)<,and 0<λ λ. Then C λ,λ can be covered by m translates of λ γ + λ λ )C. Proof If λ = λ,thenc λ,λ is a translate of λ K {0})andcanbecoveredbymtranslates of λ γ )C =λ γ )C. Now we consider the case when λ := λ 0, ) and λ =.Thereexistsasetofmpoints {ū i : i [m]} R n such that K ū i + γ K). Let z be an arbitrary point in C λ,. Then there exist ȳ K and μ [λ,] such that z = μȳ,0)+ μ)p. Lemma shows that there exists i [m]suchthat z λū i,0)+λγ + λ)c. It follows that C λ, can be covered by m translates of λγ + λ)c. When 0 < λ < λ, 4) showsthatc λ,λ is a translate of λ C λ /λ,), which can be covered by m translates of λ λ γ + λ ) C =λ γ + λ λ )C. λ λ Theorem 4 Suppose that m Z + and γ = Ɣ m K). Then Ɣ m+ C) γ. Proof Put λ =/ γ ). Then, by 4), C 0,λ can be covered by a translate of λc, andc λ, can be covered by m translates of λγ + λ)c = λc. Thus Ɣ m+ C) λ. Remark 5WhenK isaplanarconvexbody,wehavethatɣ 7 K) /. Moreover, if K is centrally symmetric, then we always have Ɣ 7 K)=/see[]). Therefore, for a convex cone C in R having a planar convex body as a base, we have Ɣ 8 C) /, an estimation which was already obtained by Chuanming Zong in [9]. It is also mentioned in [] that Ɣ 7 K) = 5/ when K is a triangle. Therefore, when T is a tetrahedron, we have Ɣ 8 T)

5 Wu and Xu Journal of Inequalities and Applications 08) 08:86 Page 5 of 9 Table Exact values of Ɣ m )cf.[, p. ]) and Ɣ m T)cf.[9, p. 559]) and consequences of Theorem 4 for Ɣ m C)withC = T m Ɣ m ) Ɣ m T) 4 Ɣ m C) /7 < /; see Table,where is a triangle and C is a cone whose base is.wehavea numerical example showing that the estimation Ɣ 8 T) /7 is not optimal either. Covering functionals of double cones Let K R n be a convex body, m be a positive integer such that Ɣ m K) <,andp, q R n+ \ R n {0} be two points such that [p, q] K {0}).Theset C = conv K {0} {p, q} ) is called a double cone. Theorem 6 Let C be a double cone defined as above. Then Ɣ m+ C) Proof Put Ɣ m K). γ = Ɣ m K) and λ = Ɣ m K). Then λ /, ). Without loss of generality we may assume that [p, q]intersectsk {0} in the origin o of R n+. By applying a suitable non-singular affine transformation if necessary, we may assume that p = e n+ and q = αe n+,whereα is a positive number. Suppose that [β, β ] [ α,].put C β β = { z C :z e n+ ) [β, β ] }. Then it is clear that C = C λ C λ)α α C λ)α. First we show that C λ λc + λ)e n+. Let z be an arbitrary point in C λ.ifz = p,thenzis clearly in λc + λ)p. In the following we assume that λ z e n+ )<.

6 Wu and Xu Journal of Inequalities and Applications 08) 08:86 Page 6 of 9 Then there exist λ, λ, λ 0, and x K {0} such that λ i =, λ λ αλ <, and z = λ x + λ p + λ q. i [] It follows that z = λ x +λ αλ )e n+ = λ x +λ αλ + λ )e n+ + λ)e n+ λ + αλ λ = λ λ On the one hand, we have x + λ αλ + λ λ + αλ λ ) p + λ)p. 5) 0 λ αλ + λ λ, which implies that λ αλ + λ λ And on the other hand, we have [0, ]. 6) 0 λ = λ λ λ + αλ, which shows that since o K {0}) λ λ + αλ x K {0}. 7) From 5), 6), and 7) it followsthat z λc + λ)p. In the second step we show that C λ)α α λc + λ)q. Similarly, we only need to consider the case when α <z e n+ ) λ)α. Inthiscase, there exist λ, λ, λ 0, and x K {0} such that λ i =, α < λ αλ α λ), and z = λ x + λ p + λ q. i [] We have z = λ x + ) λ + λ)α αλ en+ λ)αe n+ λ α = λ λα x λ ) λ)α + αλ αe n+ λ)αe n+ λα λ + α αλ λ α = λ λα λ + α λ α x + λ ) λ)α + αλ q λα + λ)q. 8)

7 Wu and Xu Journal of Inequalities and Applications 08) 08:86 Page 7 of 9 Moreover, the following inequalities hold: λ + α αλ >0, 9) 0 αλ = α λ λ ) λ + α αλ, 0) and 0 λ λ)α + αλ λα. ) Then 8) and equalities 9), 0), and ) show that z λc + λ)q. In the following we assume that {ū i : i [m]} R n is a set of points such that K ū i + γ K). Suppose that z e n+ ) [0, λ]. Then there exist λ, λ, λ 0, x K {0} such that λ i =,0 λ αλ λ, z = λ x + λ p + λ q. i [] In this situation we have z = λ x +λ αλ )e n+ = λ + αλ ) Moreover, λ λ + αλ x +λ αλ )e n+. ) 0 λ = λ λ λ + αλ λ λ + αλ [0, ], which shows that Put x := λ λ + αλ x K {0}. μ = λ + αλ. Then μ [λ, ]. From Lemma it follows that z λū i,0)+λc for some i [m]. Therefore 0 λūi,0)+λc ). C λ It remains to consider the case when z e n+ ) [ λ)α, 0]. Then there exist λ, λ, λ 0, x K {0} such that λ i =, λ)α λ αλ 0, z = λ x + λ p + λ q. i []

8 Wu and Xu Journal of Inequalities and Applications 08) 08:86 Page 8 of 9 Table Exact values of Ɣ m K )cf.[, p. ] ) and Ɣ mk )cf.[9, p. 559]) and consequences of Theorem 6 for Ɣ m C)withC = K m Ɣ m K ) Ɣ m K ) Ɣ m C) We have Clearly, z = λ x +λ αλ )e n+ = λ x λ + αλ αe n+ α αλ = α + λ αλ α x + λ + αλ α + λ αλ α q. 0 αλ = α λ λ ) α + λ αλ, which shows that x := αλ α + λ αλ x K {0}. Moreover, one can easily verify that μ = α + λ αλ α [λ,]. Again, by Lemma, z λū i,0)+λc for some i [m]. Therefore C λ) 0 λūi,0)+λc ). In Table, C is a double cone whose base is K by K p n we denote the unit ball of the Banach space lp n). Compared with exact values of Ɣ mk ), the estimations of Ɣ mc)given by Theorem 6 are optimal for m =6,7,8. 4 Conclusion Let K beaconvexbodyinr n and C be a compact convex cone in R n+ having K {0} as abase.weprovedthat Ɣ m+ C) Ɣ m K). A similar estimation is also provided for double cones. These estimations are optimal for particular pairs of m and K, are better than existing estimations, but they are not always optimal. In the authors opinion, it is interesting to do the following: provide better estimations of the covering functionals of cones and double cones, characterize convex bodies that

9 Wu and Xu Journal of Inequalities and Applications 08) 08:86 Page 9 of 9 are sufficiently close to cones or double cones via their boundary structure, and, more importantly, get precise values of Ɣ nk)whenk is an n-simplex or K n for n. Acknowledgements The authors are grateful to Chan He for her useful comments and suggestions, and we would like to thank one of the reviewers for reminding us of the connection between the covering functional and the entropy number. Funding Senlin Wu is supported by the National Natural Science Foundation of China, grant numbers 74 and Competing interests The authors declare that they have no competing interests. Authors contributions Both the authors contributed equally to this article. All authors read and approved the final manuscript. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 9 April 08 Accepted: 8 July 08 References. Boltyanski, V., Martini, H., Soltan, P.S.: Excursions into Combinatorial Geometry. Universitext. Springer, Berlin 997). Martini, H., Soltan, V.: Combinatorial problems on the illumination of convex bodies. Aequ. Math. 57 ), 5 999). Brass, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer, New York 005) 4. Bezdek, K.: Classical Topics in Discrete Geometry. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York 00) 5. Bezdek, K., Khan, M.A.: The geometry of homothetic covering and illumination. ArXiv e-prints 06) 6. Carl, B., Stephani, I.: Entropy, Compactness and the Approximation of Operators. Cambridge Tracts in Mathematics, vol. 98. Cambridge University Press, Cambridge 990) 7. Zong, C.: Functionals on the spaces of convex bodies. Acta Math. Sin. Engl. Ser. ), ) 8. Gruber, P.M.: The space of convex bodies. In: Handbook of Convex Geometry, Vols. A, B, pp North-Holland, Amsterdam 99) 9. Zong, C.: A quantitative program for Hadwiger s covering conjecture. Sci. China Math. 59), ) 0. Bezdek, K., Khan, M.A.: On the covering index of convex bodies. Aequ. Math. 905), ). Lassak, M.: Covering plane convex bodies with smaller homothetical copies. In: Intuitive Geometry, Siófok, 985. Colloq. Math. Soc. János Bolyai, vol. 48, pp. 7. North-Holland, Amsterdam 987)