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Statement

Setting as in Theorem 4.2: $1$-codimensional cylinders $C_i=B_i+H_i$ and $r$-fold packings in the sense of Definition 4.1 (p. 4); $\omega_m$ is the volume of the Euclidean unit ball $B_2^m$.

Theorem 5.1 (p. 9). Let $K$ be a convex body in $\mathbb R^d$. For $i\le N$ let $C_i=B_i+H_i$ be $1$-codimensional cylinders in $\mathbb R^d$ forming an $r$-fold packing in $K$. Then

$$\sum_{i=1}^N\operatorname{vol}_{d-1}(B_i)\le c_d\,r\max_{\dim L=d-1}\operatorname{vol}_{d-1}(P_LK), \qquad (7)$$

where $c_d=d\,\omega_d/(2\omega_{d-1})\sim\sqrt{\pi d/2}$ as $d\to\infty$.

Remark 5.2 (p. 9). The paper notes that by Theorem 4.2 one can take $c_d=1$ when $K$ is an ellipsoid, and asks for the best value of $c_d$ and for a characterization of the convex bodies satisfying (7) with $c_d$ bounded by an absolute constant.

Proof pointer

P. 9. The traces $\bar C_i\cap\operatorname{bd}K$ form an $r$-fold packing of the boundary, so their surface areas sum to at most $r$ times the surface area $s(K)$; each base has $(d-1)$-volume at most half the surface area of its trace. Cauchy's formula writes $s(K)$ as $1/\omega_{d-1}$ times the integral of $\operatorname{vol}_{d-1}(P_{u^\perp}K)$ over $S^{d-1}$, and $\lambda(S^{d-1})=d\omega_d$ bounds this by $d\omega_d/\omega_{d-1}$ times the largest hyperplane projection.

Read depth

Claims checked: Theorem 5.1 and Remark 5.2 were read clause by clause on the print, and the proof on p. 9 was followed.

Dependencies

Theorem 4.2 (for Remark 5.2 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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