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**1 - 4**of**4**### A generalization of the convex Kakeya problem

- Theoretical Informatics – 10th Latin American Symposium (LATIN 2012

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### Minimum Convex Container of Two Convex Polytopes under Translations

, 2014

"... Given two convex d-polytopes P and Q in Rd for d ≥ 3, we study the problem of bundling P and Q in a smallest convex container. More precisely, our problem asks to find a minimum convex set containing P and Q that are in contact under translations. For dimension d = 3, we present the first exact algo ..."

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Given two convex d-polytopes P and Q in Rd for d ≥ 3, we study the problem of bundling P and Q in a smallest convex container. More precisely, our problem asks to find a minimum convex set containing P and Q that are in contact under translations. For dimension d = 3, we present the first exact algorithm that runs in O(n³) time, where n denotes the number of vertices of P and Q. Our approach easily extends to any higher dimension d > 3, resulting in the first exact algorithm.

### Scandinavian Thins on Top of Cake: New and improved algorithms for stacking and packing

"... We show how to compute the smallest rectangle that can enclose any polygon, from a given set of polygons, in nearly linear time; we also present a PTAS for the problem, as well as a linear-time algorithm for the case when the polygons are rectangles themselves. We prove that finding a smallest conve ..."

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We show how to compute the smallest rectangle that can enclose any polygon, from a given set of polygons, in nearly linear time; we also present a PTAS for the problem, as well as a linear-time algorithm for the case when the polygons are rectangles themselves. We prove that finding a smallest convex polygon that encloses any of the given polygons is NP-hard, and give a PTAS for minimizing the perimeter of the convex enclosure. We also give efficient algorithms to find the smallest rectangle simultaneously enclosing a given pair of convex polygons.