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Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
9
votes
Accepted
Riemannian distance functions on the real line
[Moved from the comments/chat.]
Define $D(x)$ by
$$
\begin{align*}
D(x) =
\begin{cases}
d(0,x), & \text{ if } x\geq 0;\\
-d(x,0), & \text{ if } x< 0.
\end{cases}
\end{align*}
$$
Function $D(x)$ is n …
17
votes
Accepted
Efficiently determine if convex hull contains the unit ball
The problem is NP hard. Here is a proof sketch.
The problem is to determine if there is a point $y$ with $\|y\|=1$ outside of the convex hull of given points $x_1,\dots, x_n$. Note that such point e …