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Statement

Setting (pp. 2 and 4). A $k$-codimensional cylinder ($0<k<d$) is a set $C=B+H$ with $H$ a $k$-dimensional linear subspace of $\mathbb R^d$ and $B$ a measurable set in $E=H^\perp$; for a convex body $K$, $\operatorname{crv}_K(C)=\operatorname{vol}_{d-k}(B)/\operatorname{vol}_{d-k}(P_EK)$, with $P_E$ the orthogonal projection onto $E$. By Definition 4.1 (p. 4), cylinders $C_i=B_i+H_i$, $i\le N$, with $E_i=H_i^\perp$ and $\bar C_i=C_i\cap K$, form an $r$-fold packing in $K$ when $B_i\subset P_{E_i}K$ for every $i\le N$ and each point of $K$ belongs to at most $r$ of the interiors $\operatorname{int}(\bar C_i)$; a $1$-fold packing is a packing.

Theorem 4.2 (p. 4). Let $K$ be an ellipsoid in $\mathbb R^d$ and let $C_1,\ldots,C_N$ be $1$-codimensional cylinders in $\mathbb R^d$ forming an $r$-fold packing in $K$. Then

$$\sum_{i=1}^N\operatorname{crv}_K(C_i)\le r. \qquad (3)$$

Remark 4.3 (p. 4). If $K$ is a convex body, $d_K$ its Banach--Mazur distance to the Euclidean ball $B_2^d$, and $T$ an invertible linear map with $d_K^{-1}TB_2^d\subset K\subset TB_2^d$, then $1$-codimensional cylinders forming an $r$-fold packing in $TB_2^d$ satisfy $\sum_i\operatorname{crv}_K(C_i)\le r\,d_K^{\,d-1}$.

Proof pointer

Pp. 4--5. Since $\operatorname{crv}$ is invariant under invertible affine maps, take $K=B_2^d$. Weight $\mathbb R^d$ by the density $p(x)=(1-|x|^2)^{-1/2}$ on the open unit ball, whose integral along every chord parallel to a fixed line through $0$ is $\pi$. By Fubini the ball then has measure $\pi\omega_{d-1}$ and each truncated cylinder has measure $\pi\operatorname{vol}_{d-1}(B_i)$; the $r$-fold packing condition bounds the sum of the latter by $r$ times the former, which is (3).

Read depth

Claims checked: Definition 4.1, Theorem 4.2 and Remark 4.3 were read clause by clause on the print, and the proof on pp. 4--5 was followed.

Dependencies

None.

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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