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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 4.6 (p. 6). Let $0<k<d$ and let $K$ be a convex body in $\mathbb R^d$. Let $C_i=B_i+H_i$, $i\le N$, be $k$-codimensional cylinders in $\mathbb R^d$ forming an $r$-fold packing in $K$, and let $\bar C_i=C_i\cap K$. Assume that the $\bar C_i$ are convex bodies in $\mathbb R^d$. Then the print states

$$\sum_{i=1}^N\operatorname{crv}_K(C_i)\le r\binom dk\max_{i\le N}\frac{\max_{x\in\mathbb R^d}\operatorname{vol}_k\bigl(K\cap(x+H_i)\bigr)}{\max_{x\in\mathbb R^d}\operatorname{vol}_k\bigl(C_i\cap(x+H_i)\bigr)}.$$

The denominator as printed has $C_i$, an unbounded cylinder whose sections by the translates $x+H_i$, $x\in B_i$, are whole $k$-flats; the proof on the same page bounds each $\operatorname{crv}_K(C_i)$ with $\operatorname{vol}_k(\bar C_i\cap(x+H_i))$ in the denominator, so the bound proved has $\bar C_i=C_i\cap K$ there.

Proof pointer

Pp. 6--7. The packing condition gives $P_{E_i}\bar C_i=B_i$ with $E_i=H_i^\perp$. The Rogers--Shephard inequality (Theorem 2.1, p. 2) applied to $\bar C_i$, and its Fubini reverse (Remark 2.2, p. 2) applied to $K$, bound $\operatorname{crv}_K(C_i)$ by $\binom dk$ times $\operatorname{vol}_d(\bar C_i)/\operatorname{vol}_d(K)$ times the ratio of maximal sections. The $r$-fold packing condition gives $\sum_i\operatorname{vol}_d(\bar C_i)\le r\operatorname{vol}_d(K)$.

Theorem 2.1 (p. 2, Rogers and Shephard). For $1\le k\le d$, a convex body $K$ in $\mathbb R^d$ and a $k$-dimensional subspace $E$, $\max_{x}\operatorname{vol}_{d-k}(K\cap(x+E^\perp))\operatorname{vol}_k(P_EK)\le\binom dk\operatorname{vol}_d(K)$.

Read depth

Claims checked: Theorem 4.6 and Theorem 2.1 were read clause by clause on the print, and the proof on pp. 6--7 was followed far enough to identify the denominator it proves.

Dependencies

None in the corpus. External input: the Rogers--Shephard inequality (C. A. Rogers and G. C. Shephard, J. London Math. Soc. 33 (1958), 270--281).

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. The journal text was not compared.

Bears on

No problem page of this corpus.