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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 $r$-fold packings in the sense of Definition 4.1 (p. 4).

Theorem 4.4 (p. 6). Let $K$ be an ellipsoid in $\mathbb R^d$ and let $C_1,\ldots,C_N$ be $2$-codimensional cylinders in $\mathbb R^d$ forming an $r$-fold packing in $K$. Then

$$\sum_{i=1}^N\operatorname{crv}_K(C_i)\le r. \qquad (5)$$

Remark 4.5 (p. 6). For a convex body $K$ and an invertible linear $T$ with $d_K^{-1}TB_2^d\subset K\subset TB_2^d$, where $d_K$ is the Banach--Mazur distance from $K$ to the Euclidean ball, $2$-codimensional cylinders forming an $r$-fold packing in $TB_2^d$ satisfy $\sum_i\operatorname{crv}_K(C_i)\le r\,d_K^{\,d-2}$, inequality (6).

Proof pointer

Pp. 5--6. The proof of Theorem 4.2 is repeated with the surface measure of $S^{d-1}$ in place of the chord density, an idea the paper takes from Akopyan, Karasev and Petrov: for a plane $H$ through $0$, this measure, integrated over each translate $H+z$ with $z\in H^\perp$ and $|z|<1$, is $2\pi$, with a hint for the computation on p. 6.

Read depth

Claims checked: Theorem 4.4 and Remark 4.5 were read clause by clause on the print, and the proof sketch on pp. 5--6 was followed.

Dependencies

Theorem 4.2, whose proof is repeated.

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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