# Questions tagged [affine-geometry]

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### Existence of an open set with specific properties

I am currently reading Gerard Laumon's Faisceaux caractères, which details various constructions and properties of character sheaves on a connected reductive group $G$ over $\overline{\mathbb{F}}_p$. ...
817 views

### Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$?

Let $P$ be a convex polytope in $\mathbb{R}^d$ with $n$ vertices and $f$ facets. Let $\text{Proj}(P)$ denote the projection of $P$ into $\mathbb{R}^2$. Can $\text{Proj}(P)$ have more than $f$ facets? ...
94 views

### How does the affine Desargues theorem imply the little Desargues theorem?

Let $A$ be an (abstract) affine plane. We call $A$ a translation plane if the group of translations acts transitively on the set of points (Axiom 4a in Artin's book "Geometric algebra"). ...
210 views

### If any two triangles of equal area can be mapped via affine maps, what can we say about the geometry?

This is a cross-post. Let $(M,g)$ be a two-dimensional compact surface, endowed with a Riemannian metric. Fix $s>0$, and suppose that for any two geodesic triangles $A,B$ having area $s$, there ...
79 views

### Segre's theorem in $3$ dimensions with a "twist"

As I understand, there is a $3$-dimensional analogue of Segre's theorem stating that the maximum size of a set in ${\bf F}_q^3$ ($q$ odd) with no three points collinear is $q^2+1$. I am trying to ...
249 views

### What are some compact Hessian manifolds?

In case this is too general, here is a more specific question. Is there a hyperbolic threefold which admits a Hessian metric (hyperbolic or otherwise)? Background A Hessian manifold is a Riemannian ...
23 views

### On the perturbation of one vertex of an n-simplex so that point the given point on one of its face gets into its interior

I have a closed cube $Q_{0}=[0,1]^l$ and half-plane $H=\{(z_1,z_2,...,z_l) \in \mathbb{R}^l : z_1 + z_2 +...+ z_l > \alpha \}$ in Euclidean space $\mathbb{R}^l$ with $0<\alpha<1$. Consider ...
20 views

### How to prove continuity in topological group action of ${\rm{GA}}_a(X)$ on $T(X)$, to make ${\rm{GA}}(X)$ a topological group?

The question comes from the following paragraph of a text on geometry in the context of affine geometry (Marcel Berger et al., "Geometry I", P56-57): 2.7.1.3. If we don't want to resort to ...
93 views

### Ellipsoid in $L^p([0,1],\lambda)$ spaces?

Let us consider $L^p([0,1],\lambda)$ spaces, were $\lambda$ is simply the lebesgue measure. These are Banach spaces for $p\ge1$ (of course). It is well known that for $1\leq p < q \leq +\infty$ we ...
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 ...