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Results tagged with mg.metric-geometry
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user 13093
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
9
votes
4
answers
2k
views
Books about capacity theory
While I was studying the book Variation et Optimisation de formes by Antoine Henrot and Michel Pierre, I encountered a section about the capacity associated to the $H^1$ norm, which is defined for eve …
- 2,222
1
vote
Strengthened version of Isoperimetric inequality with n-polygon
The keyword you are looking for is "quantitative isoperimetric inequalities". The case of polygons was solved in the following paper: https://arxiv.org/abs/1402.4460
The "quantitative" term in the pap …
- 2,222
0
votes
Three circles intersecting at one point
It is straightforward to see that $A',B',C'$ are reflections of the circumcenter $O$ with respect to $BC, CA,AB$. Therefore, the center of $(AOA')$ is just the intersection of the mediatrix of $OA$ wi …
- 2,222
1
vote
Intersection point of three circles
Switching the roles of $ABC$ and $A'B'C'$, consider the circles passing through the vertices of a triangle $A,B,C$, midpoints of the opposite sides $A',B',C'$ and the circumcenter $O$.
It is straightf …
- 2,222
0
votes
A generalization of Napoleon's theorem
Notice that triangles $ACD, AEB, FCB$ are similar. Working out the ratios of the sides and the angles one can see that:
Triangles $AI_1I_3$ and $ACE$ are similar. Rotating $I_1I_3$ around $A$ with an …
- 2,222
2
votes
0
answers
409
views
Important lines in triangle - reverse problem
It is known that if three numbers $x,y,z$ are the lengths of the edges of some triangle, then there exists a triangle with medians of length $x,y,z$. Also, if $x,y,z>0$ (no condition imposed) there ex …
- 4,713
1
vote
0
answers
69
views
Integral of the square of the areas of slices of a shape
Suppose $\omega$ is a bounded shape in $\Bbb{R}^3$ and that $\{z : (x,y,z) \in \omega \}=[0,T]$ (that is, the shape is exactly contained in the band $\{z \in [0,T]\}$. If we denote by $\omega_t = \{(x …
- 9,007
1
vote
What polygons can be shrunk into themselves?
Here is my variant, a bit more geometrical.
Denote by $P$ the original polygon, and $P_\lambda$ the contracted polygon with a factor $\lambda \in (0,1)$ which lies inside $P$. Note that $P_\lambda$ i …
- 2,222
4
votes
Minimum distance between two arbitrary circles in space?
I didn't manage to solve the problem (edit: in the meantime an answer was posted which says a precise formula using radicals cannot be found), but I can post a proof that the line joining the points w …
- 2,222
2
votes
Maximum area of intersection between annulus and circle?
There is a formula for the area of the intersection of two circles of given radii in terms of the distance between the centers. The formula can be found here: http://mathworld.wolfram.com/Circle-Circl …
- 2,222