# Tagged Questions

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vote

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49 views

### vertex representation and half-space representation of a polytope [migrated]

i'm not a mathematician. So, my question may be stupid. Hope that it won't make you angry.
Is there any mathematical relationship between the vertex representation and the half-space representation ...

**2**

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**0**answers

47 views

### Intrinsic volumes of a family of convex sets $\{K_n\}$

Consider a family of convex sets $\{K_n\}$ such that $K_n \subset \mathbb{R}^n$ for each $n$. The kinds of sets one might be considering could be, for instance,
$K_n$ is the cube of side $2A$, ...

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**2**answers

283 views

### Triangle with largest perimeter in a convex region

What is the largest value of $r$ such that the following statement is always true?
"Let $C$ be a convex region with area $1$. There must exist a triangle contained in $C$ whose perimeter is at least ...

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196 views

### Unit ball of smallest volume in a Hilbert geometry

In a letter to Felix Klein published in Mathematische Annalen 1895 (see here), Hilbert generalized the Cayley-Klein model of hyperbolic geometry by defining a metric on the interior of a convex body ...

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vote

**2**answers

69 views

### Extreme points and centroid

It is well-known that the centroid of a triangle is the intersection point of its three medians. The medians happen to be area bisectors, but it seems that most (all?) other lines through the ...

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44 views

### Measure of points with small neighborhood in convex bodies

Let $K \subseteq \mathbb{R}^n$ be a "fat" convex body, i.e. one that contains a ball of radius 1. I'm interested in the following question about points $y \in K$: If you take a normally distributed $e ...

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**1**answer

103 views

### Approximation of a convex body by a contained polytope

This question deals with approximating a convex body (a compact convex set of $\mathbb{R}^d$ with non-empty interior) by convex polytopes.
For a given $\delta$, let $n_\delta$ be the number of faces ...

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votes

**1**answer

74 views

### Generalizations of Directly Similar Theorem?

There is an attractive theorem that says that if two plane figures
are directly similar, then so is any convex combination of them.
Below, $P_1$ and $P_2$ are directly similar polygons: they have
the ...

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votes

**1**answer

140 views

### Shortest curve with given convex hull

Suppose $S\subset\mathbb{R}^2$ is compact and convex. Suppose $\Gamma:[0,1]\to S$ is a continuous curve that passes through every extreme point of $S$, i.e., the convex hull of $\Gamma([0,1])$ is $S$. ...

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**1**answer

185 views

### Map from a convex polygon that increases distance

At the risk of asking an extremely stupid question, suppose that $P\subset\mathbb{R}^2$ is a convex polygon with area $1$ that contains the origin, and let $r$ denote the farthest distance between the ...

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**1**answer

95 views

### Maximizing ratio volume/diameter^n by an affinity

Suppose we have a convex compact body $D\subset \mathbb R^n$. We can try to apply affine transformation keeping the volume and decreasing the diameter of $D$.
It is clear that there is a constant ...

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131 views

### Choquet theory and Hilbert's fourth problem

The following text is an attempt to see Hilbert's fourth problem in a new light.
Definition. A pseudometric $d$ on $\mathbb{R}^n$ is called projective if whenever a point $z$ belongs to a line ...

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**1**answer

102 views

### Circumscribed ellipsoid of minimum Hilbert-Schmidt norm

Let $K\subseteq \mathbb{R}^n$ be a full-dumensional convex body. The Löwner ellipsoid of $K$ is the unique ellipsoid of smallest volume containing $K$. My question is about a related object: the ...

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229 views

### Lattice points and convex bodies

Given are two convex bodies $K, L \subset \mathbb{R}^n$ that contain the origin as an interior point. Assume the number of integer points contained in $\lambda K$ equals the number of integer points ...

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83 views

### Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between edges are rational multiples of $\pi$?

After reading these very interesting questions, I came up with another one:
Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between all pairs of edges meeting ...

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119 views

### Variations on a problem of S. Mazur

In problem 76 of the Scottish Book Mazur asked
Given a convex body $K$ in three-dimensional space and a point $o$ in its interior, consider the surface $S$ formed by all points $p$ such that the ...

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63 views

### Is the size of $\varepsilon$-nets of the Euclidean ball exponential for large $\varepsilon$

Let $X$ be the unit ball of $(\mathbb{R}^n,\|\cdot\|_2)$. A finite set $N=N(\varepsilon)$ is a $\varepsilon$-net of $X$ if every point in $X$ is at most a distance $\varepsilon$ from a point from $N$.
...

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**1**answer

174 views

### Largest convex hull of a unit length path

What is the largest area possible for the convex hull of a path of unit length lying on a plane? For what paths is that largest area attained?

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**1**answer

179 views

### n-simplex in an intersection of n balls

Consider any $n$-simplex, $n \geq 2$. For each edge $(i,j)$, consider $n$-ball $B_{ij}$ such that vertices $x_i$ and $x_j$ are antipodal on this ball. Fix a point $x_0$ in the simplex. The question: ...

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**0**answers

107 views

### Hilbert metric of a sum of cones

Suppose that $K_{1}$ and $K_{2}$ are pointed closed cones in a finite-dimensional space $V$ whose Hilbert metrics $d_{1},d_{2}$ are known. Is there a way to express the Hilbert metric of $K_{1}+K_{2}$ ...

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**1**answer

327 views

### (A question about)${}^3$ 3-dimensional convex bodies

Related to the questions mathoverflow.net question No. 137850 and mathoverflow.net question No. 39127, is there a 3-dimensional convex body other than a ball whose perpendicular projections in all ...

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vote

**1**answer

129 views

### A question concerning convex functions

Consider points $a,b,c$ (not on a line) and $x_1,...,x_n$ in $\Bbb{R}^2$. I am looking for a necessary and sufficient condition in terms of the geometric configuration of these points such that for ...

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**1**answer

221 views

### A question about a question about 3-dimensional convex bodies

For each positive integer n let E(n) denote n-dimensional Euclidean space and let the term "n-dimensional convex body" mean a compact convex subset of E(n) whose interior (with respect to E(n)) is ...

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**1**answer

89 views

### Covering a convex body with its smaller homothetic copies

Given a convex body $C\subset R^d$ and a positive real $\lambda$, any set of the form $\lambda C + x = \{ \lambda c + x \mid c\in C \}$ for some $x\in R^d$ is called a homothetic copy of $C$. The ...

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138 views

### Minkowski's convex body theorem for ellipsoids

Minkowski's theorem states that if $K\subseteq\mathbb{Z}^n$ is a convex compact set, $K=-K$, and $\mathrm{volume}(K)\geq 2^n$, then $K$ contains a nonzero integral vector.
Can this bound be improved ...

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121 views

### Hilbert metric and cross-ratio of points on simplices

Background and motivation:
Consider the cone $C\subset \mathbb{R}^d$ of vectors with non-negative components, and let $\Delta\subset C$ be the simplex of probability vectors (those for which $\sum ...

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votes

**2**answers

225 views

### Higher dimensional convex hull

Let $CH(S)$ be a convex hull of a finite set $S$ and denote the set of all the vertices of $CH(S)$ as $Vert(S)$. For a vertex $v \in Vert(S)$, it has an associated set $E(v)$ which is defined as ...

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votes

**1**answer

93 views

### Maximal cross sections of the Cartesian product of two planar domains

Let $K$ and $L$ be two two-dimensional convex bodies, and consider their Cartesian product $K\times L\subseteq \mathbb R^4$. Now let $U_\theta\subseteq \mathbb R^4$ be the two-dimensional subspace ...

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270 views

### Volume-like property to upper bound lattice points in a convex body

The following question arises in passing in a joint paper that I am working on. Let $K$ be a centrally symmetric convex body in an $n$-dimensional real vector space $V$ which contains a lattice $L$. ...

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200 views

### Can you see smoothness of the boundary of a convex body from its shadow?

Let $d \geq 3$ and suppose that $K \subset \mathbb{R}^{d}$ is a convex body (compact, convex, non-empty interior). Is the following true?
The boundary $\partial K$ is a $C^1$-manifold if and only ...

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113 views

### Almost Isodiametric Sets

Hi,
The isodiametric inequality tells us that, of all sets of diameter $r$, the one with the largest Lebesgue measure is the ball of radius $r/2$ - and this holds regardless of norm. Let $\tau(r)$ be ...

**19**

votes

**4**answers

543 views

### Do random projections (approximately) preserve convexity?

The Johnson-Lindenstrauss lemma implies that any set of $k$ points in $\mathbb{R}^d$ can be randomly projected into $d' \approx \log(k)/\epsilon^2$ dimensions such that the distances between each pair ...

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votes

**1**answer

307 views

### Detecting a hidden convex body with line probes

Imagine that, somewhere inside an origin-centered, unit-radius sphere
$S$ in $\mathbb{R}^3$,
sits a convex body $K$ of volume vol$(K)=\alpha (\frac{4}{3} \pi)$,
with $\alpha < 1$ the fraction of ...

**5**

votes

**0**answers

171 views

### Generalization of the non-existence of a monostatic planar body

Domokos, Papadopulos, and Ruina showed that there does not exist a convex planar rigid body of uniform density which has only
one orientation of stable equilibrium and one orientation of unstable ...

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votes

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117 views

### Convexified threshold of a function

Upd. The question in a nutshell: find convex set on plane which is «closest» to a given non-convex set.
It is given integrable function $0\leq f(x,y)\leq 1$ with bounded support: $f(x,y)=0$ when ...

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**3**answers

462 views

### finding the most-isolated point in a high-dimensional cube

I have a set of points {$x_1,\ldots,x_n$} located in the d-dimensional unit cube $[0,1]^d$. $n$ is about 1000 and $d$ is about 25. I'd like to find
$\max_{\omega\in [0,1]^d}\min_{i=1,\ldots,n} ...

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votes

**3**answers

345 views

### Tetrahedron angles sum to $\pi$: Bisector plane

I discovered empirically what to me is an amazing lemma
concerning face angles of a tetrahedron.
Let $\triangle abc$ be a triangle in the $xy$-plane, and $d$ the
apex of a tetrahedron with positive ...

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votes

**0**answers

139 views

### Parameterizing infinitesimal perturbations of the sphere using signed measures

Let $\mathcal K^3$ be the space of convex bodies, with some metric $\delta$, and let $B$ be the unit ball. Let us define a volume-preserving perturbation of the sphere to be a continuous (substitute ...

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votes

**1**answer

726 views

### Reference request: Ehrhart's conjecture on the geometry of numbers

Conjecture (Ehrhart). If a convex body $K \subset {\mathbb R}^n$ has its barycenter at the origin and contains no other point with integer coordinates, the volume of $K$ is less than or equal to $(n ...

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votes

**3**answers

242 views

### Expected minimum face angle of random convex polyhedron in $\mathbb{R}^3$

Let $P_n$ be a "random convex polyhedron" in $\mathbb{R}^3$ of $n$ vertices, where "random" could follow any one
of a number of models:
(1) the convex hull of $n$ points randomly and uniformly ...

**4**

votes

**1**answer

275 views

### Generalized widths and reverse Urysohn inequalities

This question is inspired by the discussion in MO questions "Local minimum from directional derivatives in the space of convex bodies" and "Bodies of constant width?" about generalized notions of ...

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**0**answers

252 views

### Local minimum from directional derivatives in the space of convex bodies

I have a function $f(K)$ defined on the space of three-dimensional convex bodies for which I want to show that the unit ball $B$ is a local minimum. I have been able to show if $K$ is not homothetic ...

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votes

**4**answers

801 views

### Delaunay triangulations and convex hulls

This is a reference request.
I have the impression that those who work in computational geometry are accustomed to the following. You have some locally finite set of sites in $\mathbb{R}^n$ and you ...

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votes

**3**answers

868 views

### Isometric (?) embedding problem.

Given a convex surface $S$ in $\mathbb{R}^3,$ you can associate to each point the length of the inward pointing normal contained inside $S.$ Call this quantity $s(x).$ Now, given a positive function ...

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votes

**1**answer

310 views

### Isometric embedding a convex cap to render its boundary planar

I would like to know if there is a polyhedral analog to this beautiful
theorem of Hong:
Theorem 11.0.1.
Any smooth positive disk $(\bar{D},g)$ with a positive geodesic
curvature along ...

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votes

**1**answer

271 views

### Angle btween Coordinate Vector and Normal Vector of Facet in a Convex Polytope, Asking for a Counterexample

Definitions
Let $\mathcal{C}$ be a convex polytope in $\mathbb{R}^{D}$ with $K$-facets
$F_{1},\ldots,F_{K}$. I denote the normal vector of the $k^\mathrm{th}$ facet as
...

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votes

**2**answers

522 views

### Do the elementary properties of mixed volume characterize it uniquely?

Background
Take 2 convex sets in $\mathbb{R}^2$, or 3 convex sets in $\mathbb{R}^3$, or generally, $n$ convex sets in $\mathbb{R}^n$. "Mixed volume" assigns to such a family $A_1, \ldots, A_n$ a ...

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votes

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471 views

### Concentration of measure for arbitrary convex bodies?

There are various "concentration-of-measure" theorems,
the best known that due to Lévy,
which is this (informally): the volume of a sphere $S^d$ in $d$ dimensions is largely
concentrated around an ...

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votes

**3**answers

745 views

### Minimum norm of convex hull

Dear all,
I am currently stuck at a problem which seems too easy to be stuck at to me...
Summary
Let $H$ be the convex hull of the points $d_1,\ldots, d_n\in \mathbb{R}^d$. How can one compute
...

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votes

**2**answers

466 views

### Do elongated convex objects all have long simple geodesics?

Let $S$ be a closed convex surface, the boundary of a compact
convex body in $\mathbb{R}^3$.
I am interested in whether there are conditions on its shape
that ensure that it supports a long, simple ...