Bezdek 2016 packing convex bodies cylinders

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lemma_6_3: Bezdek and Litvak's lemma for a convex body K in R^d with circumradius R_K: every nonnegative integrable function on K whose integral over each hyperplane section of the interior is at least Delta has integral over K at least 2 Delta R_K.

theorem_3_1: Bezdek and Litvak's r-fold covering bound: k-codimensional cylinders that cover a convex body K in R^d r times have cross-sectional volumes summing to at least r/binom(d,k), and to at least r when k = 1 and K is an ellipsoid.

theorem_4_2: Bezdek and Litvak's packing bound for an ellipsoid K in R^d: if 1-codimensional cylinders form an r-fold packing in K, their cross-sectional volumes with respect to K sum to at most r.

theorem_4_4: Bezdek and Litvak's packing bound for an ellipsoid K in R^d: if 2-codimensional cylinders form an r-fold packing in K, their cross-sectional volumes with respect to K sum to at most r.

theorem_4_6: Bezdek and Litvak's packing bound for any convex body K in R^d and any codimension k: if k-codimensional cylinders form an r-fold packing in K and each C_i cap K is a convex body, their cross-sectional volumes sum to at most r binom(d,k) times the largest ratio of maximal k-dimensional sections of K and of C_i cap K parallel to H_i.

theorem_4_8: Bezdek and Litvak's example limiting the upper bounds: for d > 3, 1 <= k < d and 0 < delta < pi/4, some k-codimensional cylinders packed in the Euclidean ball have cross-sectional volumes summing to at least c sqrt(d) (sin delta)^{2-k}/(2^{d-2}(d-k)^{3/2}).

theorem_5_1: Bezdek and Litvak's bound for any convex body K in R^d: the bases of 1-codimensional cylinders forming an r-fold packing in K have total (d-1)-volume at most c_d r times the largest (d-1)-dimensional projection of K, with c_d = d omega_d/(2 omega_{d-1}).

theorem_6_1: Bezdek and Litvak's packing counterpart of Falconer's bounds in the plane: planks forming an r-fold packing in an NS-domain K have total width at most r diam_NS(K), and the circumradius of K satisfies 2R_K <= diam_NS(K), with equality exactly when 2R_K = diam_NS(K) = diam(K).

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Bezdek, Károly and Litvak, Alexander E., Packing convex bodies by cylinders. Discrete Comput. Geom. 55 (2016), no. 3, 725--738. doi:10.1007/s00454-016-9760-z.

The paper supplies the packing counterpart of the authors' earlier covering estimates for cylinders, which arose from Bang's plank problem and his question on the base areas of cylinders covering a three-dimensional convex body. For k-codimensional cylinders C_i = B_i + H_i, E_i = H_i^perp, and the cross-sectional volume crv_K(C_i) = vol_{d-k}(B_i)/vol_{d-k}(P_{E_i} K), Theorem 3.1 extends the covering lower bound to r-fold coverings, sum crv_K(C_i) >= r/binom(d,k), and sum crv_K(C_i) >= r when k = 1 and K is an ellipsoid; its proof is left to the earlier paper. Theorems 4.2 and 4.4 show that for an ellipsoid K any r-fold packing by 1-codimensional or 2-codimensional cylinders satisfies sum crv_K(C_i) <= r. Remarks 4.3 and 4.5 bound sum crv_K(C_i) for any convex body K by r d_K^(d-1) and r d_K^(d-2), d_K the Banach-Mazur distance from K to the ball, for cylinders forming an r-fold packing in an ellipsoid T B_2^d with d_K^(-1) T B_2^d inside K inside T B_2^d. Theorem 4.6 treats an arbitrary convex body K and any codimension k, provided the truncated cylinders C_i cap K are convex bodies, through the Rogers-Shephard inequality (Theorem 2.1). Theorem 4.8 packs the ball with cylinders over caps whose total cross-sectional volume is bounded below, and Remark 4.9 compares that lower bound with Theorem 4.6's upper bound for the same cylinders. Theorem 5.1 bounds the total base volume of an r-fold packing by 1-codimensional cylinders in any convex body. Section 6 gives a packing analog of Falconer's results: Theorem 6.1 bounds the total width of an r-fold plank packing in an NS-domain K, the convex hull of a nonseparable family of disks, by r diam_NS(K), the sum of the disks' diameters times r, and proves 2R_K <= diam_NS(K) for the circumradius R_K, through Lemma 6.3 and Falconer's extremal function for the disk; the paper points to Goodman and Goodman for a different proof of that inequality.

Edition. The labels and pages on this card and its result pages are those of the arXiv version, arXiv:1507.05115v2 (21 November 2015); the journal text was not compared.

Read status: claims checked for every result page below, read clause by clause on the arXiv print; the proofs were followed as each page's Read depth states, and Theorem 3.1's proof, which lies in the earlier paper, was not read.

Source: https://arxiv.org/abs/1507.05115. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1507.05115), every other right reserved.

Bears on.

• #1121: Theorem 6.1's inequality 2R_K <= diam_NS(K), for the convex hull K of a family of closed disks that no line disjoint from all of them divides into two nonempty sets, puts the disks inside a disk whose radius is the sum of their radii, which is the problem's statement; the paper derives the inequality from Lemma 6.3 and the upper bound m(L_1^+(K)) <= diam_NS(K), its inequality (14).

Results.

• theorem_3_1 (p. 3): k-codimensional cylinders forming an r-fold covering of a convex body K in R^d, 0 < k < d, have sum crv_K(C_i) >= r/binom(d,k), and >= r when k = 1 and K is an ellipsoid.
• theorem_4_2 (p. 4): for an ellipsoid K in R^d and 1-codimensional cylinders forming an r-fold packing in K, sum crv_K(C_i) <= r.
• theorem_4_4 (p. 6): the same bound for 2-codimensional cylinders r-fold packed in an ellipsoid.
• theorem_4_6 (p. 6): for any convex body K, 0 < k < d, and k-codimensional cylinders r-fold packed in K with C_i cap K convex bodies, sum crv_K(C_i) is at most r binom(d,k) times the largest ratio, over i, of the maximal section of K by a translate of H_i to that of C_i cap K. The print has C_i in the denominator; its proof uses C_i cap K, without which the denominator would be infinite.
• theorem_4_8 (p. 7): for d > 3, 1 <= k < d and delta in (0, pi/4), some packing of the unit ball by k-codimensional cylinders has sum crv_{B_2^d}(C_i) >= c sqrt(d) (sin delta)^(2-k)/(2^(d-2) (d-k)^(3/2)), c an absolute constant.
• theorem_5_1 (p. 9): for any convex body K in R^d and 1-codimensional cylinders r-fold packed in K, sum vol_{d-1}(B_i) <= c_d r times the largest hyperplane projection of K, c_d = d omega_d/(2 omega_{d-1}).
• theorem_6_1 (p. 10): planks forming an r-fold packing in an NS-domain K in R^2 have total width at most r diam_NS(K), and 2R_K <= diam_NS(K), with equality if and only if 2R_K = diam_NS(K) = diam(K).
• lemma_6_3 (p. 11): for a convex body K in R^d with circumradius R_K, every nonnegative function whose integral over each hyperplane section of the interior is at least Delta has integral over K at least 2 Delta R_K.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.