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- W2952407455 abstract "For two planar convex bodies, $C$ and $D$, consider a packing $S$ of $n$ positive homothets of $C$ contained in $D$. We estimate the total perimeter of the bodies in $S$, denoted ${rm per}(S)$, in terms of ${rm per}(D)$ and $n$. When all homothets of $C$ touch the boundary of the container $D$, we show that either ${rm per}(S)=O(log n)$ or ${rm per}(S)=O(1)$, depending on how $C$ and $D$ fit together, and these bounds are the best possible apart from the constant factors. Specifically, we establish an optimal bound ${rm per}(S)=O(log n)$ unless $D$ is a convex polygon and every side of $D$ is parallel to a corresponding segment on the boundary of $C$ (for short, $D$ is emph{parallel to} $C$). When $D$ is parallel to $C$ but the homothets of $C$ may lie anywhere in $D$, we show that ${rm per}(S)=O((1+{rm esc}(S)) log n/log log n)$, where ${rm esc}(S)$ denotes the total distance of the bodies in $S$ from the boundary of $D$. Apart from the constant factor, this bound is also the best possible." @default.
- W2952407455 created "2019-06-27" @default.
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- W2952407455 date "2014-09-10" @default.
- W2952407455 modified "2023-10-16" @default.
- W2952407455 title "On the total perimeter of homothetic convex bodies in a convex container" @default.
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- W2952407455 doi "https://doi.org/10.1007/s13366-014-0219-1" @default.
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