The Covering Index of Convex Bodies

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The Covering Index of Convex Bodies Centre for Computational and Discrete Geometry Department of Mathematics & Statistics, University of Calgary uary 12, 2015

Covering by homothets and illumination Let E d denote the d-dimensional Euclidean space. A d-dimensional convex body K is a compact convex subset of E d with nonempty interior. A homothet of K is a set of the form λk + x, where λ is a nonzero real number and x E d. Conjecture 1 (Covering Conjecture) K can be covered by 2 d of its smaller positive homothets and 2 d homothets are needed only if K is an affine d-cube. The illumination number I (K) of K is the smallest n for which the boundary of K can be illuminated by n points/directions. Boltyanski (1960) showed that I (K) = n if and only if the smallest number of smaller positive homothets of K that cover K is n. Conjecture 2 (Illumination Conjecture) I (K) 2 d, and I (K) = 2 d only if K is an affine d-cube.

The illumination parameter Question: How economically can we illuminate K by a few nearby point sources? Bezdek (1992) introduced the illumination parameter ill(k) of a d-dimensional o-symmetric convex body K. { } ill(k) = inf p i K : {p i } illuminates K, p i E d, i where x K = inf{λ > 0 : x λk} is the Minkowski functional of the symmetric convex body K. Far away light sources are penalized. I (K) ill(k), for o-symmetric convex bodies. Bezdek (1992, 2006), Bezdek, Boroczky and Kiss (2006), Kiss and De Wet (2010) and others calculated ill(k) for various symmetric convex bodies. uhammad Ali Khan (and Károly Bezdek)

The covering parameter Question: How economically can we cover K by a few small homothets? Swanepoel (2005) defined the covering parameter of a d-dimensional convex body. { 1 C(K) = inf : K 1 λ i i i (λ i K + t i ), 0 < λ i < 1, t i E d }. Large homothets are penalized. ill(k) 2C(K), if K is o-symmetric. C(K) = O(2 d d 2 ln d), if K is o-symmetric. C(K) = O(4 d d 3/2 ln d), in general. Let C d denote a d-dimensional cube, then C(C d ) = 2 d+1. uhammad Ali Khan (and Károly Bezdek)

Two ingredients: the Banach-Mazur distance The (multiplicative) Banach-Mazur distance between d-dimensional convex bodies K and L is d BM (K, L) = inf {δ 1 : a K, b L, L b T (K a) δ(l b)}, where the infimum is taken over all invertible linear operators T : E d E d. Denote by K d the (compact) space of d-dimensional convex bodies under Banach-Mazur distance. d BM is a multiplicative metric on K d. Used to study affine invariant functionals of convex bodies. Can think of d BM as distance between d-dimensional real Banach spaces.

Two ingredients: γ m (K) Define γ m (K) to be the minimal homothety ratio required to cover K by m positive homothets. γ m (K) = inf { λ > 0 : K } m (λk + t i ), t i E d, i = 1,..., m. i=1 Originally, introduced by Lassak (1986). Zong (2010) reintroduced it as a functional on K d and proved it to be uniformly continuous. We observed that in fact, γ m (K) d BM (K, L)γ m (L), for any K, L K d.

The covering index Let K d denote the space of d-dimensional convex bodies. We define the covering index of K as { } m coin(k) = inf 1 γ m (K) : γ m(k) 1/2, m N. (1) Intuitively, coin(k) measures how well K can be covered by a relatively small number of positive homothets all corresponding to the same relatively small homothety ratio.

Why γ m (K) 1/2? 1) Rogers (1963), Verger-Gaugry (2005), O Rourke (2012) and others investigated the minimum number of homothets of ratio 1/2 or less needed to cover a d-dimensional ball. 2) Easy to obtain optimizers. 3) Define m f m (K) = 1 γ m (K), if γ m(k) 1/2,, otherwise. Then coin(k) = inf {f m (K) : m N}. For any K, L K d and m N such that γ m (K) 1/2 and γ m (L) 1/2, f m (K) d BM (K, L)f m (L), f m (K) d BM(K, L) 2d BM (K, L) 1 f m(l). (2) Relations (2) don t work without restricting the homothety ratios.

Relationship with other quantities Proposition 1 For any o-symmetric d-dimensional convex body K, vein(k) ill(k) 2C(K) 2 coin(k), and in general I (K) C(K) coin(k). Here vein(k) = inf { i I p i K : K conv { p i E d : i I }} denotes the vertex index of a symmetric convex body K defined by Bezdek and Litvak (2007).

Upper bounds Proposition 2 (Rogers-type bounds) Given K K d, d 2, we have 2 2d+1 d(ln d + ln ln d + 5) = O(4 d d ln d), K o-symmetric, ( ) coin(k) < 2d 2 d+1 d(ln d + ln ln d + 5) = O(8 d d ln d), otherwise. d The proof uses Rogers estimate of the translative covering density of K, Rogers-Shephard inequality on the volume of the difference body K K = K + ( K), and the Rogers-Zong inequality on the minimum number of translates of a convex body L that cover K.

Monotonicity Lemma 1 Let j < m be positive integers. Then for any d-dimensional convex body K the inequality f j (K) > f m (K) implies m < f j (K). This shows that the covering index of any convex body can be obtained by calculating a finite minimum, rather than the infimum of an infinite set. In particular, if f j (K) < for some j, then coin(k) = min {f m (K) : m < f j (K)}.

A simple example The figure shows that an affine regular convex hexagon H can be covered by 6 half-sized homothets. Thus coin(h) f 6 (H) 12. uary 12, 2015 12

A simple example Any half-sized homothet of H can cover at the most one-sixth of the boundary of H. Therefore, γ m (H) > 1/2, for m = 1,..., 5 and f 6 (H) = 12. Thus, coin(h) = inf{f m (H) : 6 m < 12} 12. If f m (H) < 12, for some 7 m 11, then γ m (H) < 12 m 12, and by the definition of covering, m vol(γ m (H)H) = mγ m (H) 2 vol(h) vol(h). Therefore, ( ) 12 m 2 m > 1, 12 which is impossible for 8 m 11. This only leaves the case m = 7. But Lassak (1987) showed that γ 7 (H) = 1/2 and as a result, f m (H) = 14. We conclude that coin(h) = 12.

Direct vector sums Let E d = L 1 L n be a decomposition of E d into the direct vector sum of its linear subspaces L i and let K i L i be convex bodies. Denote the direct sum of K 1,..., K n by K 1 K n. Boltyanski and Martini (2007) showed that I (K 1... K n ) n j=1 I (K j), but that the equality does not hold in general. No general formula exists for I (K 1... K n ) in terms of illumination of the K j s. Let us denote by N λ (K), the minimum number of homothets of K of homothety ratio 0 < λ 1 needed to cover K. Then { m coin(k) = inf 1 γ m (K) : γ m(k) 1 } 2, m N N λ (K) = inf λ 1 1 λ. 2

Direct vector sums Theorem 1 Let E d = L 1 L n be a decomposition of E d into the direct vector sum of its linear subspaces L i and let K i L i be convex bodies such that coin(k i ) = f mi (K i ), i = 1,..., n, and Γ = max{γ mi (K i ) : 1 i n}. Then max{coin(k i ) : 1 i n} n i=1 coin(k 1 K n ) = inf N λ(k i ) λ 1 1 λ 2 n i=1 N Γ(K i ) n i=1 m n i 1 Γ 1 Γ < coin(k i ). Moreover, the first two upper bounds in (3) are tight. i=1 (3)

1-codimensional cylinders Let K E d E d E 1 = E d+1 be a d-dimensional convex body and l E 1 E d E 1 = E d+1 denote a line segment that can be optimally covered (in the sense of coin) by two homothets of homothety ratio 1/2. We say that the d + 1-dimensional convex body K l E d+1 is a 1-codimensional cylinder. Theorem 2 For any 1-codimensional cylinder K l, the first two upper bounds in (3) become equalities and coin(k l) = 4N 1/2 (K).

Minkowski sums Let d denote the d-simplex. Theorem 3 Let the convex body K be the vector sum of the convex bodies K 1,..., K n in E d, i.e., let K = K 1 + + K n, such that coin(k i ) = f mi (K i ), i = 1,..., n, and Γ = max{γ mi (K i ) : 1 i n}. Then n i=1 coin(k) = coin(k 1 + +K n ) N Γ(K i ) n i=1 m n i 1 Γ 1 Γ < coin(k i ). Moreover, equality in (4) does not hold in general. (E.g., H = 2 + ( 2 ), but coin( 2 ) = coin(h) = 12.) i=1 (4)

Continuity Theorem 4 Let K d m := { K K d : γ m (K) 1/2 }. (i) For any K, L K d m, f m (K) d BM (K, L)f m (L), f m (K) d BM(K, L) 2d BM (K, L) 1 f m(l). (ii) The functional f m : Km d R is uniformly continuous for all d and m. (iii) Define I K = {i : γ i (K) 1/2} = {i : K Ki d }, for any d-dimensional convex body K. If I L I K, for some K, L K d, then (but not in general) coin(k) d BM (K, L) coin(l), coin(k) 2d BM(K, L) 1 d BM (K, L) coin(l). (iv) The functional coin : K d R is lower semicontinuous for all d.

Optimizers Theorem 5 Let B d and C d denote the d-dimensional ball and d-dimensional cube, respectively. (i) For any K K d, coin(c d ) = 2 d+1 coin(k) and so affine d-cubes minimize the covering index in all dimensions. (ii) If K is a planar convex body then coin(k) coin(b 2 ) = 14. (iii) If K l is a 1-codimensional cylinder in K 3, then coin(k l) 28, that is B 2 l maximizes coin among 1-codimensional 3-dimensional cylinders.

Another example: coin(b d ), d 3 Recently, O Rourke (2012) raised the question as to what is the minimum number of homothets of homothety ratio 1/2 needed to cover B 3. Using spherical cap coverings, Wynn (2012) showed this number to be 21. In fact, if the homothety ratio is decreased to 0.49439, we can still cover B 3 by 21 homothets. Therefore, coin(b 3 ) 41.53398.... Verger-Gaugry (2005) showed that in any dimension d 2 one can cover a ball of radius 1/2 < r 1 with O((2r) d 1 d 3/2 ln d) balls of radius 1/2. This shows that coin(b d ) = O(2 d d 3/2 ln d).

Known values/estimates K m γ m (K) coin(k) l 2 1/2 4 H 6 1/2 12 2 6 1/2 12 B 2 7 1/2 14 B 3 21 0.49439 41.53398... B d O(2 d d 3/2 ln d) 1/2 O(2 d d 3/2 ln d) C d 2 d 1/2 2 d+1 H l 12 1/2 24 2 l 12 1/2 24 B 2 l 14 1/2 28

Open questions Problem 1 (Upper semicontinuity) Either prove that coin is upper semicontinuous or construct a counterexample. Since circles maximize coin in the plane, it is reasonable to propose the following. Problem 2 (Maximizers) For any d-dimensional convex body K, prove or disprove that coin(k) coin(b d ) holds. An affirmative answer would considerably improve the known general (Rogers-type) upper bound on the illumination number from O(4 d d ln d) to O(2 d d 3/2 log d).

References K. Bezdek and M. A. Khan, The covering index of convex bodies, preprint. Centre for Computational and Discrete Geometry http://math.ucalgary.ca/ccdg.

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