All Questions
Tagged with affine-geometry convex-geometry
9
questions
1
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317
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Geometry in $\mathbb{R}^n$: angle between projections of a rectangle
Consider a hyper rectangle $R$ in $\mathbb{R}^n$ defined by $|x_i|\leq M_i$ for all $i\leq n$.
Consider a linear affine subspace $L$ of dimension $1\leq k <n$ such that $L\cap R\neq \emptyset$.
For ...
CommunityBot
- 1
2
votes
1
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On convex solids with all plane sections affine congruent
Question: How many (classes of) convex 3D solids are there such that all non-degenerate planar sections of the solid are mutually affine congruent?
Further question: Same as above with 'projective' ...
- 166
7
votes
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Separating a lattice simplex from a lattice polytope
Let $P\subset\mathbb{R}^n$ be a convex lattice polytope.
Do there always exist a lattice simplex $\Delta\subset P$ and an affine hyperplane $H\subset\mathbb{R}^n$ separating $\Delta$ from the convex ...
- 8,852
5
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Cohn-Vossen rigidity theorem in hyperbolic space
There is the following rigidity theorem of Cohn-Vossen as stated on p. 86 of these lecture notes: http://www.math.brown.edu/~deigen/chern.pdf
Any isometry between two closed smooth convex surfaces in ...
- 10.3k
4
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Maximin diameter of a transformed convex figure
You are given a compact convex figure in the plane, $C$.
Your goal is to transform $C$ to a different figure $C'$ with the smallest possible diameter, as long as:
The trasformation from $C$ to $C'$ ...
CommunityBot
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4
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2
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478
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How to show the two convex bodies are affinely isomorphic?
This problem comes from the response of the author of papers.
Consider two convex bodies $A$ and $B$:
$$A= \{X\in \mathcal{S}^4 : \operatorname{tr}(X) = 1, X\succeq 0 \}$$
$$B = \operatorname{...
- 103k
1
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154
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Affine-regular hexagon in convex body
An affine-regular $n$-gon is a non-degenerate affine image of the regular $n$-gon. It seems to be a standard fact in combinatorial geometry that inside every convex compact set $K\subseteq \mathbb R^2$...
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Osculating ellipsoids
Let $K$ be a given smooth, origin-symmetric, strictly convex body in $n$ dimensional euclidean space. At every point $x$ on the boundary of $K$ there exists an origin-symmetric ellipsoid $E_x$ that ...
- 4,744
5
votes
2
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367
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Diameter-area ratio for affine tranformations.
Consider an convex plane figure $F$. How to prove that there is an affine transformation $a$ such that $\sqrt{3}$ diameter$(a(F))^2\leq 4$ area$(a(F))$?
I found only one reference, to "Über einige ...
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