# Questions tagged [mg.metric-geometry]

Euclidean, hyperbolic, discrete, convex, coarse geometry, comparisons in Riemannian geometry, symmetric spaces.

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### find a parametric equation for the that is perpendicular to the graph of the given equation at the given point: z=x3-xy2 , (-1,1,0) [closed]

i am still a beginner in this and don't know how to do it so please can you help? find a parametric equation for the line that is perpendicular to the graph of the given equation at the given point: z=...
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### Normal from a point to paraboloid [closed]

Prove that the normals from the point ( α,β,γ ) to the paraboloid $x^2/a^2 + y^2/b^2 = 2z$ lie on the cone $α/(x-α) + β/(y-β) + (a^2 -b^2)/(z-γ) = 0$
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### Which polytopes can be deformed while keeping their edge-lengths?

Let $P\subset\Bbb R^d$ be a convex polytope (a convex hull of finitely many points). Lets call it flexible, if it can be continuously deformed while keeping its combinatorial type, and keeping its ...
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### Can you measure the degree of uniformity of a 2d shape?

Is there a calculation that could take the points that make of the outline of a 2 dimensional shape and provide a numeric evaluation representative of the uniformity or symmetry of the shape. Such as ...
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### Generalized quadrangles and their connection to prime powers

Generalized quadrangles are a commonly know geometric structures. A generalized quadrangle is an incidence structure $(P,B,I)$, with $I \subseteq P \times B$ an incidence relation, satisfying certain ...
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### Minimal data required to determine a convex polytope

Let $P\subset \Bbb R^d$ be a convex polytope. Suppose that I know its combinatorial type (aka. the face-lattice), the length $\ell_i$ of each edge, and the distance $r_i$ of each vertex from the ...
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### Trade-off between covering number, ball radius and diameter of $d$-dimensional shapes

Given any $d$-dimensional shape $X$ in the Euclidean space, let $\ell(X)$ be the length of the longest line segment connecting two points of $X$. How can we prove the following statement? There exists ...
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### Neighbor count in sphere packing in N dimensions

So I'm really interested in building a mathematical model for how powerful computer chips could be given extra spatial dimensions. Obviously this is a squishy problem, since "computer chips" ...
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### Can prolates overlap more easily than oblates?

Context: When modeling anisotropic particles, the two common types of shapes of interest are cylindrical and disk-like particles. For simplicity let us say we model these as prolates and oblates ...
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### Klein geometry associated to a degenerate conic

In his study of non-euclidean geometries, Felix Klein considers the group of projective transformations acting on the real projective plane whose extensions to the complex projective plane preserve a ...
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### Can we define geodesic in the space of compactly supported functions?

From Wikepedia, the definition of geodesic is stated as: A curve $\gamma: I\to M$ from an interval $I$ of the reals to the metric space $M$ is a geodesic if there is a constant $v\geq 0$ such that ...
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### Two questions on counterexamples to Borsuk's conjecture and ball-packings

In 1933 Karol Borsuk conjectured the following Can every bounded subset $E$ of $\mathbb{R}^d$ be partitioned into $(d+1)$ sets, each of which has a smaller diameter than $E$? Whilst new to this ...
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### Is Sydler's theorem concerning Dehn invariants constructive?

Sydler proved something of a converse to Dehn's negative resolution of Hilbert's 3rd problem. To quote Wikipedia, Sydler showed that "every two Euclidean polyhedra with the same volumes and Dehn ...
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### Applications of the co-area formula

Kirchheim  generalized the classical area formula to the case of Lipschitz mappings into metric spaces. Ths paper is well known and widely cited. The area formula is a special case of the co-area ...
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### number of integer points inside a triangle and its area

Let $T$ be a triangle in $\mathbb{R}^2$ defined by $y = \alpha x$, $y = \beta$ and $x = \gamma$ where $\alpha, \beta, \gamma \in \mathbb{R}_{>0}$. I am interested in obtaining an estimate for the ...
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### If the volume-ratio of an inscribed convex set to the circumscribing convex set is rational, can anything of consequence be further deduced?

Say, one has two $n$-dimensional convex sets $A$ and $B$, with $B$ being inscribed in the strictly larger set $A$. ($A$ and $B$ have at least one boundary point in common. $B$ “fits snugly” in $A$ ...
Let $ABC$ denotes a triangle and $p(ABC)$ denotes its perimeter. We say two points $O_1$ and $O_2$ inside this triangle are perimeter points if there are points $a$, $b$ and $c$ on the sides $BC$, $AC$...
Can projection of convex sets onto convex sets be non-convex yet connected? If so is there any necessary and sufficient conditions? Can projection of $n$ dimensional convex sets in $\mathbb R^{n'}$ ...