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2 votes

To place copies of a planar convex region such that number of 'contacts' among them is maximized

The left shape below has $3$ contacts (circled) "between pairs of units" and hull area $> 3$, while the right shape has $2$ contacts and area $3$. So minimizing the hull area does not ...
Joseph O'Rourke's user avatar
3 votes
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Characterization of convexity by connectedness of hyperplane sections

Suppose $n\geqslant 2$. Let $S$ be the boundary of $\tfrac12$-neighborhood of the unit $n$-sphere in $\mathbb{R}^{2{\cdot}n+1}$. (Note that $S$ is homeomorphic to $\mathbb{S}^n\times\mathbb{S}^n$.) ...
Anton Petrunin's user avatar

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