# Questions tagged [geometry]

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### Axioms of betweenness

In Axiomatic Geometry, groups of axioms for points and lines are introduced. The second of these groups are called the axioms of betweenness. Given three points $A,B,C$ we postulate a relation $A*B*C$,...
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### How can the same polytope have three different volumes? [closed]

I'm quite new to geometry and I came across the idea that the same convex polytope can have at least three different volumes. Consider the permutohedron, formed by the convex hull of the n! points ...
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### Minimum volume of intersection between two high-dim $\ell^1$-balls

Let $B_1$ and $B_2$ be two balls with the same radius, in $\mathbb R^n$ with the $\ell^1$ norm. The distance between the centers of $B_1$ and $B_2$ is $d(B_1, B_2)$. Is there any deterministic method ...
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### Identity involving dot product of solid angle and gradient [closed]

How to prove following for $n\geq0$ ? $$\int_{4\pi}d\vec{\Omega}(\vec{\Omega}\cdot\vec{\nabla})^{2n}f(\vec{r})=\frac{4\pi}{2n+1}\nabla^{2n}f(\vec{r})$$ Where, at any point $\vec{r}$, the $\vec{\Omega}$...
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### State-of-the-art geometry book? [closed]

For my best friend's birthday, I am looking for a geometry book. He's currently doing his math PhD and is really fond of geometry, especially hyperbolic or higher-dimensional ones, also interested in (...
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### Does there exist concept of angles and sides in Non-archimedian field geometry? [closed]

Question about Non-Euclidean Geometry, particularly about Non-archimedian Geometry: I am studying and trying to understand about Non-Euclidean Geometry for my future purpose. An example of Non-...
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### Is there a dense subset on closed Jordan curve $C$ which its points make intersections under certain rotations?

Is it true that for any given closed Jordan curve of $C \subset \mathbb{R}^2$ there is a dense subset $A$ such that for every point $p\in A$ we have the following property: If we rotate $C$ around $p$...
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### Functional inequality under mean curvature flow

Let $\Sigma$ be a hypersurface in $\mathbb R^n$ and $\Sigma_t$ be a variation of $\Sigma$ under the mean curvature flow under an extra condition that ${\rm vol}_{n-1}(\Sigma)={\rm vol}_{n-1}(\Sigma_t)$...
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### Need help maximizing distances to nearest neighbor in a cylinder

I have a cylinder and I want to maximize the number of points in the cylinder such that the distances to the nearest neighbors are maximally spaced. How do I find out how many points I can have so ...
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### geometrically infinite ends of hyperbolic 3 manifolds

Let $M$ be a hyperbolic 3-manifold with finitely generated fundamental group. Assume $E$ is a geometrically infinite end (not of geometrically finite type, i.e. the convex core can not be separated of ...
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### Find structure geometry of $A_1, A_2,…,A_n$ such that $\prod_{i<j} A_iA_j$ is maximum

In any triangle we have the well-known inequality: $$\sin{A}\sin{B}\sin{C} \le \frac{3\sqrt{3}}{8} (1)$$ Signification of inequality (1): Let three points $A, B, C$ lie on a circle then $AB.BC.CA$ ...
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### Optimization of points on a plane

Suppose we have $n$ points on a plane. Let $D$ be the sum of the squares of all the pairwise distances between the points. Let $A$ be the area of the convex hull. What is the minimum possible value of ...
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### Equidistribution of Brillouin zones

Answering the question about Limiting shape for Brillouin zones Victor Kleptsyn proved that $N$th Brillouin zone is very close to a circle of radius $c\sqrt N$ (you can find all necessary definitions ...
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### Limiting shape for Brillouin zones

Is it true that the limiting shape for Brillouin zones (for any lattice) is a circle? You can find the definition and the step by step construction of Brillouin zones here. This picture is taken from ...
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### Existence of polytope

Does there exist a polytope in dimensional d consisting of $k>d+1$ faces satisfying that every d faces intersect? I tried 3 dimensional cases, and it seems negative. But is it all negative for any ...
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### Zoll Flat Finsler tori and convex bodies on a starry night

The starry night. The "celestial sphere" is given by set of non-zero vectors in $\mathbb{R}^n$ modulo positive dilations (i.e., $v \equiv w$ if $v = \lambda w$ for some $\lambda > 0$) and the "...