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Statement

Setting (pp. 10--11). For a convex body $K$ in $\mathbb R^d$, $\mathcal L^+(K)$ is the set of nonnegative functions on $K$ that are Lebesgue integrable over $K$. For $u\in S^{d-1}$ and $s\ge0$, $H(s,u)=\{x:\langle x,u\rangle=s\}$, and for a hyperplane $H(s,u)$ meeting $\operatorname{int}(K)$ the sectional integral $F(f,s,u)$ is the integral of $f$ over $H(s,u)\cap\operatorname{int}(K)$ with respect to $(d-1)$-dimensional Lebesgue measure on $H(s,u)$. For $\Delta>0$, $\mathcal L_\Delta^+(K)$ is the set of $f\in\mathcal L^+(K)$ with $F(f,s,u)\ge\Delta$ for every $H(s,u)$ meeting $\operatorname{int}(K)$, and $m(\mathcal L_\Delta^+(K))$ is the infimum of $\int_Kf(x)\,dx$ over $f\in\mathcal L_\Delta^+(K)$. The circumradius $R_K$ is the radius of the smallest Euclidean ball containing $K$.

Lemma 6.3 (p. 11). If $K$ is a convex body with circumradius $R_K$ in $\mathbb R^d$, then $m(\mathcal L_\Delta^+(K))\ge2\Delta R_K$.

Proof pointer

Pp. 11--12. Translate so that the $f$-weighted centroid of $K$ is the origin and take $u$ with $h_K(u)=R\ge R_K$, where $R$ is the least radius of a ball about the origin containing $K$ and $h_K$ is the support function. The vanishing first moment in the direction $u$, with $F\ge\Delta$, makes the moments $\int tF(f,t,u)\,dt$ over $[0,h_K(u)]$ and over $[0,h_K(-u)]$ each at least $\frac12\Delta R_K^2$. A one-dimensional extremal bound, (12), then gives each half of $\int_Kf$ at least $\Delta R_K$.

Read depth

Claims checked: the definitions and Lemma 6.3 were read clause by clause on the print, and the proof on pp. 11--12 was followed; the infimum (12), which the paper says one can check, was not rederived.

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.

Bears on

• Problem 1121: the lemma is the step of Theorem 6.1 that turns the paper's upper bound on $m(\mathcal L_1^+(K))$ into the circumradius inequality $2R_K\le\operatorname{diam}_{NS}(K)$.