The convex-geometry tag has no wiki summary.

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**1**answer

353 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

399 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 ...

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vote

**2**answers

464 views

### Extreme points of transportation polytope

I'm interested in $n \times m$ joint probability tables with prescribed row and column marginals. Such tables form a convex set known as the transportation polytope. What are the extreme points of ...

**2**

votes

**2**answers

554 views

### Finding smallest ellipsoid that circumscribes over intersection of two ellipsoids that do not have common center

Does this already exist in literature? The closest Ive been able to find is circumscribe intersection of two ellipsoids with a common center by W. Kahan ...

**2**

votes

**1**answer

291 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
...

**12**

votes

**1**answer

658 views

### Optimal 8-vertex isoperimetric polyhedron?

I know from Marcel Berger's
Geometry Revealed:
A Jacob's Ladder to Modern Higher Geometry
(p.531)
that it is not yet established which polyhedron in $\mathbb{R}^3$ on 8 vertices achieves the optimal ...

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vote

**0**answers

270 views

### Computing the intersection of dual affine subspaces

Suppose we have a convex function , $\phi(x): R^d \to R$. It is well known that the Legendre transform of $\phi$ is also a convex function, and can (loosely) be thought of as the dual or derivative ...

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**3**answers

833 views

### Draw a Random Line Through a Voronoi Tessellation, What is the Average Number of Voronoi Cell the Line Intersects?

Update: problem reformulation
Following the advice in comments, I now restate my problem using Voronoi
tessellation.
Given a unit hypercube $H_n=\{(x_1,\ldots,x_n)\in \mathbb{R}^n: 0\leq x_i\leq
...

**3**

votes

**1**answer

218 views

### Sobolev type inequalities involving affine metric

Let $\mathcal{M}$ be a compact smooth hypersurface in $\mathbb{R}^{n}$ with affine metric $h$ and affine volume measure $d\mu_{h}$. If
$$\int_{\mathcal{M}}|\nabla f|^2_{h}d\mu_{h}$$ is small what can ...

**14**

votes

**2**answers

594 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 ...

**15**

votes

**1**answer

444 views

### Convex bodies with constant maximal section function in odd dimensions

In 1970 or so, Klee asked if a convex body in $\mathbb R^n$ ($n\ge 3$) whose maximal sections by hyperplanes in all directions have the same volume must be a ball. The counterexample in $\mathbb R^4$ ...

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votes

**0**answers

155 views

### Maximising the volume of a parallelotope in a cone with a fixed vertex

Let $C\subset\mathbb{R}^n$ be a cone with vertex at the origin, aperture $\theta$ and height $h$. Since a cone is a convex region in $\mathbb{R}^n$ we know there is a parallelotope $P$, completely ...

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votes

**2**answers

536 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 ...

**5**

votes

**2**answers

323 views

### Distributing points with respect to a concave function

Suppose I have a concave function defined on the unit interval such that $f(0) = f(1) = 0$ and $\int_0^1 f(t) dt = \alpha$, where $\alpha$ is "small" (say $0.01$ or thereabouts). Say I distribute $n$ ...

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votes

**0**answers

201 views

### How do control the boundary regularity of the Legendre transformation domain from a convex function

Let f(x) be a strongly convex smooth function (its Hessian matrix is positive definite) defined in a convex domain D, introduce the Legendre transformation
$$x=(x_1,...,x_n)\rightarrow ...

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votes

**3**answers

718 views

### Area-preserving map between rectangles and fat polygons

Are there any well-known continuous area-preserving maps between fat convex polygons and fat rectangles? Specifically, given a fat convex polygon $C$, is there a natural way to choose a fat rectangle ...

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votes

**2**answers

1k views

### The cone of positive semidefinite matrices is self-dual? (reference needed)

I'm seeking a reference for the following fact.
The cone of positive semidefinite matrices is self-dual (a.k.a. self-polar).
This result is relatively easy to prove, has been known for a long ...

**7**

votes

**1**answer

2k views

### Geometric meaning of derivatives of the curvature

Let $\Gamma$ be a sufficiently smooth, closed, convex curve in the plane parametrized by arclength $s$. Further assume that $\kappa>0$ and that the second derivative of $\kappa$ with respect to $s$ ...

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votes

**4**answers

770 views

### Is there a midsphere theorem for 4-polytopes?

The (remarkable) midsphere theorem says that each combinatorial
type of convex polyhedron may be realized by one all of whose edges are
tangent to a sphere
(and the realization is unique if the center ...

**3**

votes

**3**answers

840 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
...

**2**

votes

**1**answer

357 views

### A min-max formula for depth of the origin in a convex set

Depth is defined as the distance to the boundary of a set, i.e., $\operatorname{depth}(x, C) = \operatorname{dist}(x, \mathbb{R}^n \backslash C)$.
Let $C$ be a convex set that contains the origin. I ...

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votes

**2**answers

585 views

### Helly theorem + Nerve

Consider nerve $\mathcal N$ of a finite set of convex sets in $\mathbb R^n$.
Helly theorem says that $\mathcal N$ is completely determined by its $n$-skeleton, say $\mathcal N_n$.
It seems that not ...

**3**

votes

**1**answer

375 views

### minimal maximal ellipsoids

Suppose $K$ is a centrally symmetric, strictly convex body in $\mathbb{R}^2$. Let denote the curvature and the support function of $\partial K$, boundary of $K$, respectively with $\kappa$ and $s$. If ...

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votes

**1**answer

861 views

### Equations for an algebraic gömböc

A gömböc is a $3-$dimensional convex body (having uniform density) which has exactly one stable and one instable equilbrium position (see http://en.wikipedia.org/wiki/G%C3%B6mb%C3%B6c).
Such a convex ...

**3**

votes

**0**answers

219 views

### Maximum of a function on $d-$dimensional convex compact sets

Let $\mathcal C_d$ denote the set of all $d-$dimensional convex compact subsets with
barycenter at the origin of the $d$-dimensional Euclidean space $\mathbb E^d$. Given an
element $C\in\mathcal C_d$ ...

**8**

votes

**1**answer

585 views

### volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?

We denote by $\otimes_{\epsilon}$ the injective Banach tensor product.
Which is the asymptotic volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?

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votes

**0**answers

195 views

### Maximum surface area among convex subsets of the unit sphere of a given volume

The following problem is listed in Steven Lay's "Convex Sets and Their Applications" (1982) as unsolved (paraphrased):
Let $B$ be the unit ball in $\mathbb{R}^3$ and $0 < V < \pi$. Define ...

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votes

**2**answers

244 views

### volume of the projected body

Suppose $K$ is an $n$-dimensional $C^2$ convex body in $\mathbb{R}^{n+1}$. We choose two distinct directions $z_0, z_1\in\mathbb{S^{n}}$. If $P_1$ and $P_2$ are the corresponding hyperplanes($z_0\perp ...

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votes

**4**answers

2k views

### Polynomial roots and convexity

A couple of years ago, I came up with the following question, to which I have no
answer to this day. I have asked a few people about this, most of my teachers and some
friends, but noone had ever ...

**8**

votes

**3**answers

1k views

### Listing lattice points in a simplex

Let $n \in \mathbf{Z}_{\geq 1}$. Is there an algorithm which, given a simplex $\Delta \subset \mathbf{R}^n$ specified as the convex hull of $v_0,\dots,v_n \in \mathbf{Z}^n$, computes the set $\Delta ...

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**0**answers

331 views

### How aspherical can a Gömböc be?

A Gömböc is a homogeneous massive convex solid that can rest on a horizontal plane in just two positions of equilibrium under gravity: one stable and the other unstable. How small a proportion of the ...

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**0**answers

949 views

### Two-convexity ⇒ Lefschetz?

Assume that
$\Omega$ is an open simply connected set in $\mathbb R^n$
(two-convexity) if 3 faces of a 3-simplex belong to $\Omega$ then whole simplex in $\Omega$.
Is it true that any component of ...

**3**

votes

**2**answers

387 views

### Submanifolds lying on the boundary of a convex domain

Let $M$ be a submanifold of $\mathbb R^n$. Call $M$ locally convex if locally $M$ is contained in the boundary of a convex domain of $\mathbb R^n$.
Is there any known condition that is equivalent to ...

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votes

**4**answers

1k views

### Analogy of Parseval identity for Legendre Transform ?

Parseval's identity states that the sum of squares of coefficients of the Fourier transform of a function equals the integral of the square of the function, or
$$ \sum_{-\infty}^{\infty} |c_n|^2 = ...

**12**

votes

**2**answers

641 views

### Point on a line nearest a point in Banach space

I have a Banach space geometry question (a curiosity-driven spin-off from a research topic). Given a point $x$ on the unit sphere of a Banach space and a vector $y\ne 0$, there is a multiple $t_0y$ of ...

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votes

**3**answers

547 views

### sharper minkowski bound

If we want to bound the norm of the smallest ideal which generates a nontrivial ideal class, is there a better bound than Minkowski's bound?
(Note that Minkowski's bound is to guarantee something ...

**8**

votes

**2**answers

497 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 ...

**2**

votes

**2**answers

1k views

### Finding points inside innermost convex hull [closed]

Given a set of points $S$ on the Euclidean plane, Onion Peeling determines the nested set $H$ of convex hulls on $S$. Define an analytical formula on $S$ which produces a point, not necessarily in ...

**1**

vote

**1**answer

295 views

### How do maximum norms relatively change in Euclidean translations

Let $Q$ be the cube $[-1,1]^{3}$ and $\pi$ be a plane in $\mathbb{R}^{3}$
that contains the origin but doesn't contain any vertex of $Q$. Suppose that $A$ is an invertible
linear transformation from ...

**1**

vote

**1**answer

195 views

### crookedness of convex curves (milnor)

hello,
I'm currently reading On the Total Curvature of Knots and am trying to understand one of the lemmas in it (3.3)
A closed polygon $P$ in $H^2$ is convex if and only if for every $b$ either ...

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vote

**1**answer

268 views

### The number of hyperplanes determining the integer points of a polyhedron

This question is inspired by this one.
Let $P \subset \mathbb R_{+}^n$ be a convex polyhedron whose complement in $\mathbb R_{+}^n$ has finite volume. Let $Int(P) = P \cap \mathbb N^n$. (For ...

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votes

**4**answers

2k views

### Ellipse naturally associated with a polygon

My colleagues and I have stumbled onto a way to associate an ellipse, or equivalently a positive definite symmetric matrix, to a polygon that is different from other better known ways. We want to know ...

**1**

vote

**2**answers

662 views

### Is there always a parallelogram cross-section of parallelepiped contained in the smallest box

Let $M$ be a centered parallelepiped, the intersection of $M$ and any plane $P$ that passes through the origin is a parallelogram or hexagon. Each parallelogram or hexagon has a cubic box that is the ...

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vote

**0**answers

387 views

### Does the dual Banach space $B(\ell^\infty)$ have weak* normal structure?

Let $K$ be a bounded closed convex subset of a Banach space $E$. A point $x$ in $K$ is called a diametral point if
$$
\sup_{y\in K} \|x-y\|={\rm diam}(K).
$$
where ${\rm diam}(K)$ denotes the ...

**21**

votes

**2**answers

1k views

### The logic of convex sets

Let me start with Helly's theorem: Let $A_1$, $A_2$, ..., $A_{n+2}$ be $n+2$ convex subsets of $\mathbb R^n$. If any $n+1$ of these subsets intersect (this means: have nonempty intersection), the so ...

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votes

**2**answers

991 views

### Are the Platonic solids shadows of 4-polytopes?

Say that a 3D shadow of a 4-polytope is a parallel projection to 3-space, not necessarily orthogonal to that 3-space (that would make it an orthogonal projection).
I am wondering if each of the five ...

**3**

votes

**1**answer

357 views

### A conjecture of parallelogram inside convex and central symmetric curve

Assume Q is a convex central symmetric curve, whose area is $\displaystyle S$. The area of the maximum parallelogram inside Q is $\displaystyle S'$.
How to prove the conjecture that $\displaystyle ...

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votes

**1**answer

195 views

### Finding a point that lies in a majority of polytopes

Suppose I have $k$ $n$-dimensional polytopes $P_1,\ldots,P_k$, each explicitly specified as the intersection of a collection of hyperplanes. If there was a point $p \in \mathbb{R}^n$ that lay in the ...

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votes

**0**answers

117 views

### Quantifying that near a point on a smooth hypersurface, it looks like a tangent hyperplane

Suppose $C$ is a smoothly bounded convex body (*) in $\mathbb{R}^d$, and $p$ is a point on the boundary of $C$. Let $r>0$ and let $B(r)$ denote the ball of radius $r$ centered at $p$.
Is it ...

**0**

votes

**3**answers

535 views

### a different algebra/representation for convex sets

Hi,
I was dealing with finding a feasible region for a set of norm inequalities and the feasible region is convex. The question is not about how to find the feasible region but how to represent the ...