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Statement

Setting (pp. 2--3). For $0<k<d$ a $k$-codimensional cylinder is a set $C=B+H$, where $H$ is a $k$-dimensional linear subspace of $\mathbb R^d$ and $B$ is a measurable set in $E=H^\perp$. For a convex body $K$ (a compact convex set with nonempty interior) the cross-sectional volume of $C$ with respect to $K$ is

$$\operatorname{crv}_K(C)=\frac{\operatorname{vol}_{d-k}(C\cap E)}{\operatorname{vol}_{d-k}(P_EK)}=\frac{\operatorname{vol}_{d-k}(B)}{\operatorname{vol}_{d-k}(P_EK)},$$

where $P_E$ is the orthogonal projection onto $E$. Sets $L_1,\ldots,L_N$ form an $r$-fold covering of $K$ when every point of $K$ belongs to at least $r$ of them.

Theorem 3.1 (p. 3). Let $K$ be a convex body in $\mathbb R^d$ and $0<k<d$, and let $C_1,\ldots,C_N$ be $k$-codimensional cylinders in $\mathbb R^d$ forming an $r$-fold covering of $K$. Then

$$\sum_{i=1}^N\operatorname{crv}_K(C_i)\ge\frac{r}{\binom dk}.$$

Moreover, if $k=1$ and $K$ is an ellipsoid, then $\sum_{i=1}^N\operatorname{crv}_K(C_i)\ge r$.

The case $r=1$ is the authors' earlier covering bound, recalled on p. 2 as inequality (1) and, for $k=1$ and an ellipsoid, on p. 3 as inequality (2), both from their 2009 paper (the paper's reference [BL]). The paper notes (p. 3) that $k=d-1$ is the affine plank problem of Bang, where (1) gives the lower bound $1/d$.

Proof pointer

No proof is printed. The paper says (p. 3) that the theorem follows by slightly modifying the proofs of Theorem 1 and Remark 2 of [BL] (K. Bezdek and A. E. Litvak, Covering convex bodies by cylinders and lattice points by flats, J. Geom. Anal. 19 (2009), 233--243).

Read depth

Claims checked: the definitions and Theorem 3.1 were read clause by clause on the print. The proof lives in [BL], which was not read.

Dependencies

None in the corpus. External input: the covering estimates of [BL].

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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