## New answers tagged convex-geometry

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 ...

3
votes

Accepted

### 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 $\mathbb{S}^n\times\mathbb{S}^n$.)
...

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