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Statement

Setting as in Theorem 4.2: $k$-codimensional cylinders $C=B+H$, cross-sectional volume $\operatorname{crv}_K(C)=\operatorname{vol}_{d-k}(B)/\operatorname{vol}_{d-k}(P_{H^\perp}K)$ (p. 2), and packings in the sense of Definition 4.1 (p. 4). $B_2^d$ is the Euclidean unit ball and $\omega_m$ the volume of $B_2^m$.

Theorem 4.8 (p. 7). Let $d>3$, $1\le k<d$ and $\delta\in(0,\pi/4)$. There exist $k$-codimensional cylinders $C_1,\ldots,C_N$ forming a packing in $B_2^d$ with

$$\sum_{i=1}^N\operatorname{crv}_{B_2^d}(C_i)=\frac1{\omega_{d-k}}\sum_{i=1}^N\operatorname{vol}_{d-k}(B_i)\ge\frac{c\sqrt d\,(\sin\delta)^{2-k}}{2^{d-2}(d-k)^{3/2}},$$

where $c$ is an absolute positive constant.

Remark 4.9 (pp. 7--8). For the cylinders of the proof, whose truncations are the convex hulls of spherical caps of geodesic radius $\delta$, the upper bound of Theorem 4.6 becomes $\binom dk(\sin\delta)^{-k}$, so in this example the upper and lower bounds differ by a factor of order $C(d,k)(\sin\delta)^{-2}$.

Proof pointer

P. 8. Take a maximal $2\delta$-separated set $\{x_i\}$ on $S^{d-1}$ in geodesic distance. Over each $x_i$ build a cylinder whose base is the solid cap of geodesic radius $\delta$ about $x_i$ in a fixed $(d-k)$-dimensional subspace through $x_i$; its truncation by the ball is the convex hull of the spherical cap $S(x_i,\delta)$, so the cylinders form a packing. Maximality makes the caps $S(x_i,2\delta)$ cover the sphere, which bounds $N$ below by the reciprocal of a cap measure. Lemma 4.7 (p. 7), $\delta(\sin\delta)^n/(e(n+1))\le\int_{\pi/2-\delta}^{\pi/2}(\cos t)^n\,dt\le\delta(\sin\delta)^n$ for $\delta\in(0,\pi/2)$ and $n\ge1$, and estimates for $\omega_m$ give the bound.

Read depth

Claims checked: Lemma 4.7, Theorem 4.8 and Remark 4.9 were read clause by clause on the print, and the proof on p. 8 was followed in outline; its final constant estimate was not rechecked.

Dependencies

Theorem 4.6 (for Remark 4.9 only).

Source. K. Bezdek and A. E. Litvak, Packing convex bodies by cylinders, Discrete Comput. Geom. 55 (2016), no. 3, 725--738, doi:10.1007/s00454-016-9760-z; labels and pages are those of arXiv:1507.05115v2 (21 November 2015), as the source card records.

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