The convex-geometry tag has no wiki summary.

**6**

votes

**0**answers

119 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 ...

**0**

votes

**2**answers

375 views

### Is minimum of convex envelope the same as minimum of the original function?

Hello everyone my question is:
$Question:$ Consider a function $f:X \rightarrow \mathbf R$ where $X$ is a convex subset of $\mathbf{R}^n$. The convex envelope of $f$ over $X$ is defined as the ...

**7**

votes

**2**answers

271 views

### Duality between extremal points and extremal maps

Suppose I have a convex set $C\subset\mathbb{R}^n$ such that $0\in C$ and every Cauchy sequence in $C$ converges in $C$, but $C$ need not be bounded. (Actually I want unbounded $C$). Consider the set
...

**19**

votes

**4**answers

639 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 ...

**11**

votes

**1**answer

335 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

**5**answers

590 views

### Finding an axis-aligned ellipsoid of minimal volume which contains a given ellipsoid

A friend asked me to post the following question. He's not an MO user and felt it would be better received if asked by someone who was already known to the community. This is not my area, but I'll do ...

**5**

votes

**0**answers

187 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 ...

**3**

votes

**2**answers

426 views

### The (Sigma) Algebra of Convex Sets

This is a question-by-proxy for a colleague from computer science. I'm sure many people here are already aware that convex decomposition forms an important sub-field of both computational geometry and ...

**0**

votes

**0**answers

179 views

### A linear program related question

Dear all, recently, I encountered the following problem. It is closely related to the order of growth for UMD constants of all $n$-dimensional Banach lattice.
Let $\alpha^k \in (\alpha_1^k, ...

**2**

votes

**1**answer

140 views

### coarser than triangulations “almost partitions” into simplices

The total space $T$ of an embedded into $\mathbb{R}^n$ pure $n$-dimensional simplicial complex (in other words, the union of finitely many $n$-dimensional compact convex polytopes) sometimes admits an ...

**2**

votes

**0**answers

66 views

### Affine endomaps and isometries of convex bodies

Let $B$ stand for a compact convex body in a Hilbert space $H$.
Suppose that a finite group $G=\{g_1,\ldots,g_n\}$ acts, via affine maps, on $B$.
Write $B'$ for the image of $B$ in $B^n$ by the ...

**4**

votes

**0**answers

131 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 ...

**1**

vote

**1**answer

671 views

### proving the interior of a dual cone is the set of vectors whose inner product is strictly positive on the cone

Apologies for posting such a simple question to mathoverflow. I've have been stuck trying to solve this problem for some time and have posted this same query to math.stackexchange (but have received ...

**4**

votes

**0**answers

217 views

### Smoothness of the convolution of a singular measure with itself

Let $\gamma:I\subset\mathbb{R}\rightarrow\mathbb{R}^2, s\mapsto \gamma(s)$, denote the arclength parametrization of a smooth, convex curve $\Gamma:=\gamma(I)\subset\mathbb{R}^2$. Equip $\Gamma$ with ...

**2**

votes

**0**answers

213 views

### Convexity and probability

Problem instance: A closed convex body $B\subset {\Bbb R}^n$ of volume 1; a point $p\in B$; and a real number $v\in(0,1)$.
Objective: Find the probability $P(B,v,p)$ that $p\in B'$, for $B'$ a ...

**2**

votes

**1**answer

311 views

### Quermassintegrals as mean curvature integrals

It is well-known that the quermassintegrals $W_{i}$ of a convex body $K\subset \mathbb{R}^{n}$ can be written as a mean curvature integral
$$ W_{i}(K)=n^{-1}\int_{\partial K}H_{i-1} ...

**11**

votes

**3**answers

489 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} ...

**3**

votes

**3**answers

521 views

### How to find the minimum number of hyperplanes to define a convex hull?

I have the following problem:
I have a convex hull $\Omega$ defined by a set of n-dimensional hyperplanes $S = [(n_1,d_1), (n_2,d_2),...,(n_k,d_k)]$ such that a point $p \in \Omega$ if $n_i^T p \geq ...

**7**

votes

**2**answers

293 views

### Continuity of barycentre in Hausdorff metric

Let $K_1$, $K_2$ be two convex compact sets in $\mathbb{R}^d$, and $p_1,p_2$ be their barycenters. Is it true that the distance between $p_1$ and $p_2$ does not exceed a Hausdorff distance between ...

**2**

votes

**2**answers

412 views

### Finite imensional subspaces of $L^\infty.$

This is the question I had meant to ask when I asked this question.: Is there a concise characterization of finite dimensional subspaces of $L^\infty?$ (that's what the discussion with @fedja was ...

**5**

votes

**3**answers

302 views

### Finite dimensional subspaces of $L^1.$

This question is motivated by my discussion (via comments) with @fedja regarding this earlier question. In any case the question is whether there is any concise characterization of finite dimensional ...

**5**

votes

**3**answers

416 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 ...

**2**

votes

**0**answers

146 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 ...

**0**

votes

**1**answer

343 views

### On the number of lines of given points

Hi all, I have a question Concerning Beck's theorem. I have read it from http://en.wikipedia.org/wiki/Beck%27s_theorem and I have two questions :
I suppose Beck's theorem doesn't hold when instead ...

**4**

votes

**2**answers

525 views

### Area ratio of a minimum bounding rectangle of a convex polygon

Take a convex polygon $P$ in the plane, and find its minimum area bounding rectangle, $R$. I'm interested in the ratio of the area of $R$ to the area of $P$. The ratio has a minimum of $1$ for ...

**9**

votes

**3**answers

1k views

### Area Enclosed by the Convex Hull of a Set of Random Points

Consider $n$ points generated randomly and uniformly on a unit square. What is the expected value of the area (as a function of $n$) enclosed by the convex hull of the set of points?

**3**

votes

**1**answer

171 views

### Realisation of convex polygons with an interior point from combinatorial data

A convex polygon $P$ having an interior point $A$ in generic position (not on any line defined by two vertices) can be encoded combinatorially by associating to every vertex $v$ of $P$ the unique
edge ...

**0**

votes

**0**answers

89 views

### centrally symmetric neighborly polytopes.

Let $P$ is centrally symmetric $k$ neighborly d-polytope. Let $ P^* $ is polar( also dual) of $ P $. Consider a polytope $ Q^* $ , $ Q^* \subset P^* $( not necessarily a face of $P^*$). By duality
...

**2**

votes

**1**answer

396 views

### faces of a polytope

Let $a_1,a_2,..,a_n\in\mathbb{R}^m$, $n>m$ and the points are in general position meaning, no hyperplane contains more than $m$ points. Define a polyhedron $$P=\{x\in\mathbb{R}^m:a_1'x\leq0, ...

**3**

votes

**1**answer

168 views

### Flattening a corner in a convex $d$-polytope (into $d-1$ dimensions, without overlap)?

I'm interested in the following question, which seems to be assumed all over the place (at least for 3 dimensions) in convex geometry, and which I cannot find a proof of.
Suppose we have a corner ...

**2**

votes

**2**answers

563 views

### Cone over the Join of two topological spaces

Suppose $X$ and $Y$ are topological spaces. Let's define the join $X\ast Y$ as the quotient space $X\times Y\times [0,1]/\sim$, where $\sim$ is the equivalence relation generated by ...

**3**

votes

**3**answers

281 views

### Vertex-transitive polytopes in any dimension with any number of vertices?

Given positive integers $d$ and $v$ with $v \geq d+1$, does there always exist a (convex) vertex-transitive $d$-polytope with $v$ vertices? It seems that the answer should be "obviously" true, but I ...

**9**

votes

**1**answer

776 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 ...

**7**

votes

**3**answers

264 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

331 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 ...

**5**

votes

**0**answers

268 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 ...

**4**

votes

**0**answers

138 views

### Relationship between sequential compactness of a convex set and its extremal points

Suppose that $X$ is a compact convex subset of a topological vector space. Suppose also that the extremal points of $X$ have the additional property that any sequence $x_n$ of extremal points has a ...

**4**

votes

**4**answers

1k 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 ...

**6**

votes

**3**answers

287 views

### Large geodesically convex subsets of tori

Let $X=\mathbb T^d=\mathbb R^d/\mathbb Z^d$ and let $E$ be a proper open subset of $X$. We say $E$ is geodesically convex if for any $x,y\in E$ the shortest geodesic connecting $x$ and $y$ lies in ...

**10**

votes

**3**answers

964 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 ...

**24**

votes

**3**answers

695 views

### Largest possible volume of the convex hull of a curve of unit length

What is the largest possible volume of the convex hull of an open/closed curve of unit length in $\mathbb{R}^3$?

**17**

votes

**2**answers

803 views

### Placing points on a sphere so that no 3 lie close to the same plane

Motivation
I am working with arbitrary parallelopiped tilings given by projection from a higher dimensional space. The collection of tiles, and some properties of the higher dimensional space are ...

**12**

votes

**2**answers

223 views

### Smallest containing simplex

Let $V_n$ be the least real number such that for every convex subset of $\mathbb{R}^n$ with hypervolume $1$ there is a containing simplex with hypervolume $V_n$.
What is known about $V_n$? Is there a ...

**1**

vote

**1**answer

361 views

### Convex Polygon - Splitting into Two Congruent Pieces

Dear All,
I have convex polygon (expressed by points in cartesian coordinate system). I am looking for a solution to splitting into two congruent pieces. Is there any way to to estimate the points ...

**6**

votes

**2**answers

378 views

### Complexity of detecting a convex body in $\mathbb{R}^n$?

Let $K_0$ be a bounded convex set in $\mathbb{R}^n$ within which lie two sets $K_1$ and $K_2$. $K_0,K_1,K_2$ have nonempty interior. Assume that,
$K_1\cup K_2=K_0$ and $K_1\cap K_2=\emptyset$.
The ...

**1**

vote

**1**answer

638 views

### Applications of ham sandwich type results. References? A general principle?

Lately there has been a lot of interest on applications of the ham sandwich theorem and related results. There is a bunch of lecture notes and surveys that touch upon the subject. I dont know of any ...

**4**

votes

**2**answers

632 views

### gradient of convex functions

Hello. Can somebody help me with the following question that I have thought over for quite some time, to no avail?
Let $f$ be a smooth function (class $\mathrm{C}^{\infty}$), $f:\mathbb{R}^n ...

**8**

votes

**4**answers

851 views

### Applications of Hilbert's metric

Among the fascinating constructions in mathematics is the Hilbert metric on a bounded convex subset of ${\mathbb R}^n$.
Where, within mathematics, is it used ? I know at least a proof of the ...

**5**

votes

**1**answer

350 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 ...

**0**

votes

**0**answers

398 views

### Upper and Lower Bounds of the Total Surface Area of Convex Polytopes that Partition a Hypercube

Let $C$ be a hypercube in $\mathbb{R}^D$ with edge length of $L$. Let $\mathcal{P}_1,\ldots,\mathcal{P}_K$ be $K$ convex polytopes that partition $C$. Let $S_k$ be the surface area of the polytope ...