Recently I learnt that $$\DeclareMathOperator{\diam}{diam}\DeclareMathOperator{\per}{per}\DeclareMathOperator{\area}{area} \inf\frac{\diam(C)(\per(C)-2\diam(C))}{\area(C)}=0$$ where the infimum is taken over all plane convex bodies $C$ (say, with non-zero area). In other words, there is no non-tr...

For the quotient space $G=G_0/G_1$, knowing the homotopy groups of $G_0$ and $G_1$, one can determine homotopy groups from the long exact sequence $$ ... \to \pi_n(G_1) \to \pi_n(G_0) \to \pi_n(G_0/G_1) \to \pi_{n-1}(G_1) \to \pi_{n-1}(G_0) \to \pi_{n-1}(G_0/G_1) \to .... $$ But in pract...

As the image presented below, the reddish point set is totally separated from the blueish one and the greenish one, while the blueish point set is quite mixed with the greenish one. A number of point sets on the plane. Each point set takes up a simply connected domain concave or convex. The poi...

a classical results by M. Inaba et al. in "Applications of Weighted Voronoi Diagrams and Randomization to Variance-Based k-CLustering" (Theorem 3) says The number of Voronoi partitions of $n$ points by the Euclidean Voronoi diagram generated by $k$ points in $d$-dimensional space is $\mathc...