Questions tagged [convex-geometry]

A branch of geometry dealing with convex sets and functions. Polytopes, convex bodies, discrete geometry, linear programming, antimatroids, ...

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21 views

Is the set $\operatorname{Unif}(0,\frac{1}{n})$ for odd and even $n$ a 2-alternating capacity?

Let $\Omega$ be a complete metrizable space $\mathscr A$ its Borel $\sigma$-algebra and $\mathscr M$ the set of all probability measures on $\Omega$. Every non-empty subset $\mathscr P \subset \...
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1answer
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Naming convention: looking for better terminology for “centrally symmetric smooth strictly convex bodies”

I have recently found myself researching a certain type of convex body in $\mathbb{R}^2$, namely centrally symmetric smooth strictly convex bodies. Instead of repeating such a sentence repetitively I ...
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Upper bound $\tau_C := \int_{\|x\| \le 1}(vol(C \cap (x + C))/vol(C))dx$ for a convex body $C \subseteq \mathbb R^n$, by reducing to a ball

Let $C$ be a convex body in $\mathbb R^n$, i.e a bounded convex subset of $\mathbb R^n$ which has nonempty interior, and which is (A) open, or (B) closed (I'm not sure one makes more sense; choose the ...
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What do you call $\operatorname{diam} (A)^d/\mathcal{L}^d (A)$ for $A \subseteq \mathbb{R}^d$ convex?

If $A \subseteq \mathbb{R}^d$ is convex, is there a more or less established name for the quantity $$\operatorname{diam} (A)^d/\mathcal{L}^d (A),$$ where $\mathcal{L}^d (A)$ is the Lebesgue measure of ...
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Separability of Minkowski Sum of well-behaved sets

Let $A$ and $C$ be non-empty simply connected and connected subsets of $\mathbb{R}^k$ and suppose that $C$ is convex. Then is the Minkowski sum $A+C$ separable?
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Polytopes with large dihedral angles

The regular $d$-simplex has dihedral angle $\arccos(1/d)<90^\circ$, and the $d$-cube has dihedral angle exactly $90^\circ$. The maximal dihedral angle of a prism over a $(d-1)$-simplex is also $90^\...
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2answers
370 views

Is Minkowski sum of boundary convex again?

Consider a closed, bounded and convex set $C \subset \mathbb{R}^{2}$ and denote its boundary with $\partial C$. It is very well-known that the Minkowski sum of two convex sets is convex again. What ...
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1answer
104 views

Is the intrinsic volume always positive for maximum dimension?

The intrinsic volume functions on $\mathbb{R}^d$ known from the Steiner formula and Hadwiger's Theorem can be extended to the domain of definable sets of an o-minimal structure $\text{Def}(\mathbb{R}^...
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81 views

Given a finite set of points, does there exist a linear function pass through a point and strictly below the other points for all the points?

I guess my question is a follow up question of this one: usul, Existence of a strictly convex function interpolating given gradients and values, version: 2019-04-13. In usul's question, the answer ...
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1answer
243 views

Equivalence of σ-convex hull and closed convex hull

Let $X$ be a locally convex topological space, and let $K \subset X$ be a compact set. Recalling that the standard convex hull is defined as $$\text{co}(K) = \Big\{ \sum_{i=1}^n a_i x_i : a_i \geq 0,\,...
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27 views

Extended-value subgradients

(I am not an expert in convex analysis, as may become clear.) Let $R$ be the extended reals, $R = \mathbb{R} \cup \{\pm \infty\}$. Standard texts define the subgradient of a convex function $f: \...
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1answer
29 views

On comparing planar convex regions of equal perimeter and area

Definitions: The Hausdorff distance between two point sets is the greatest of all the distances from a point in one set to the closest point in the other set. Given two planar convex regions $C_1$ ...
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1answer
199 views

Minimum area of the convex hull of the union of a parallelogram and a triangle

This question is somewhat dual to my previously stated question about Maximum area of the intersection of a parallelogram and a triangle, where the triangle and parallelogram each is assumed to be of ...
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2answers
285 views

On 'fair bisectors' of planar convex regions

Definitions (https://www.ias.ac.in/article/fulltext/pmsc/122/03/0459-0467): Given a planar convex region $C$ (could be smooth or polygonal), an area bisector of $C$ is any line that partitions $C$ ...
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Is Steiner symmetrization “Turing complete”?

This question stems from intuition so it is a little soft. It concerns performing computation using transformations on sets. The idea is that a rearrangement like Steiner symmetrization might be "...
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1answer
107 views

Every convex set is of locally finite perimeter

I need to prove that every convex subset of $\mathbb{R}^n$ is of locally finite perimeter. $E$ is of locally finite perimeter if there exists a vector-valued Radon measure $\mu_E$ s.t. the Gauss ...
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Is there a name for a “convex hull with holes”?

If I have a (solid) 3d object, is there a name for the object created from it by taking the convex hull and subtracting from it all points that are on a straight line between any two points on the ...
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1answer
99 views

Hessian matrix and its positiveness

Assume that $\Sigma^2$ is a closed surface in $\mathbb{R}^3$ defined by the equation $\rho(x)=1$, where $\rho$ is some smooth function so that $\nabla \rho\neq 0$. Let $A=H(\rho)$ be the Hessian ...
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3answers
445 views

Number of regions formed by $n$ points in general position

Given $n$ points in $\mathbb{R}^d$ in general position, where $n\geq d+1$. For every $d$ points, form the hyperplane defined by these $d$ points. These hyperplanes cut $\mathbb{R}^d$ into several ...
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30 views

Is there a term for a not-necessarily-convex set whose non-extreme points can be expressed as a linear combination of two other points in the set?

This question was asked on Math.SE here, but received no replies after several months. So I have posted it here, though with somewhat revised structuring of the question. Let $V$ be a real vector ...
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1answer
96 views

Flatness directions of the operator norm

It is known that the standard operator norm $\|\cdot\|_2$ over ${\bf M}_n({\mathbb R})$ is very flat, as is any operator norm (= subordinated norm) actually. The set of extremal points of the unit ...
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90 views

Quotient Banach space whose dual map sends the ball onto a given convex subset

Let $X$ be a Banach space and let $A$ be a closed, convex and balanced subset of $B_{X^{*}}$ (where $B_{X^{*}}$ denotes the closed unit ball of the dual $X^{*}$). Is there a closed subspace $M$ of $X$ ...
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1answer
249 views

A conjecture (or theorem?) on unit vectors in a Euclidean space

I have heard (if I am not mistaken) that there exists the following conjecture (or theorem?). Let $u_1,\dots,u_n$ be unit vectors in an $n$-dimensional Euclidean vector space. Then there exists ...
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Deformations that flatten small curvature

I'm trying to show that any 3-dimensional polyhedron with many vertices can be mildly deformed so that its vertices are no longer convexly independent. I suspect it suffices to look at a vertex with ...
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2answers
695 views

Reference to a conjecture on unit vectors in Euclidean space

I have heard that there exists the following conjecture (if I am not mistaken). Let $u_1,\dots,u_n$ be unit vectors in an $n$-dimensional Euclidean vector space. Then there exists another unit vector ...
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77 views

Isometries between two convex bodies [closed]

Let $A,B$ be two convex compact subsets with non-empty interior in a Euclidean space $\mathbb{R}^n$. Let $f\colon A\to B$ be a bijective isometry between them. Does there exist an isometry $F\colon ...
3
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1answer
74 views

Affine equivalence of Coxeter permutahedra?

Suppose that $W=\langle s_1,\ldots, s_d\mid (s_is_j)^{m_{ij}}=e\rangle$ is a finite reflection group and consider its standard $d$-dimensional geometric realization (i.e., the Tits representation) $\...
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58 views

Linearly independent support vectors of a convex set

Let $\Omega\subset\mathbb{R}^n$ a compact strictly convex set containing $0$ in its interior and let $k\leq n$. Given a vector $x\neq 0$ in $\mathbb{R}^n$ a supporting vector $\xi_x$ in the ...
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25 views

Convex combination of semi-algebraic sets

Suppose we are given two semi-algebraic sets $S_1$ and $S_2$. Define $$ S=\{s=ps_1+(1-p)s_2: s_1\in S_1, s_2\in S_2, 0\leq p\leq 1.\} $$ $S$ is semi-algebraic. Can we bound the degree of $S$? If we ...
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63 views

For a manifold of positive curvature, can we lower bound the distance between unit normals?

Suppose $M \subset \mathbb R^d$ is a $C^2$ $(d-1)$-manifold. In particular I am interested in when $M$ is the set $\{x \in \mathbb R^d: f(x) \le 1\}$ for some $C^2$ function $f:\mathbb R^d \to \...
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1answer
342 views

Why is a convex variety called convex?

Let $X$ be a smooth projective variety. By Definition 24.4.2 in the 2003 book Mirror Symmetry, $X$ is called convex if $h^1\left( \Sigma, f^*T_X \right) = 0$ for every genus zero stable map $f:\Sigma \...
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1answer
106 views

Number of orthants intersected by a convex hull

I'm trying to figure out the following problem: Let $x_1,\ldots,x_k\in\mathbb{R}^n$ be some points for some $k<n$. Let $\mbox{conv} (x_1,\ldots,x_k)$ be their convex hull. I'm looking for a tight (...
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94 views

How are the $L^2$ and $\sup$ norms related on the space of strongly convex functions?

Given a convex compact set $X \subset \mathbb R^d$ with interior containing the orign let $V$ be the space of all smooth functions $f: X \to \mathbb R$ with the properties: The function is strongly ...
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0answers
104 views

On intrinsic volumes

Let $\Gamma$ be a convex polytope in $\mathbb R^n$. The $k$-th intrinsic volume of $\Gamma$ is the number $$ \text{v}_k(\Gamma)=\sum_{\Delta\in{\mathcal B}(\Gamma,k)}\text{vol}_k(\Delta)\psi_\Gamma(\...
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218 views

Open convex hull of a closed set

Let $X$ be a closed set in a Euclidean space of finite dimension and suppose that its convex hull $H$ is open. I can prove that, in this case, $H$ is a Cartesian product of a line with an open convex ...
3
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1answer
100 views

Convex sets with analytic boundary, using angles to parametrize boundary

Suppose that $D$ is a bounded open convex subset of $\mathbb{R}^2$ with analytic boundary. You can parametrize the boundary of $D$ using the angles of the support lines at each point, but it isn't ...
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1answer
61 views

Closed on generic set implies closed set whole set [closed]

Assume that $f:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ is a continuous on a set $A$. Let $B \subset A$ be a generic set in $\mathbb{R}^{2}$ i.e, the countable intersection of the open and dense ...
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1answer
141 views

Convex sets on the discrete Heisenberg group

I'm interested in whether the finitely-generated discrete Heisenberg group admits a notion of "convex set". Below a formalization of what I need from the convex sets, in particular they should all be ...
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0answers
40 views

Essential rays in fan structure

Let $|\Sigma|$ be the underlying set of some fan $\Sigma$ in $\mathbb{R}^n$. It is well known that in general there is no coarsest fan structure on $|\Sigma|$. However, there may be some special rays ...
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100 views

On convex regions containing (and contained within) a given triangle

Given an arbitrary triangle T. How does one find the convex region C_M of largest area containing T such that T is also the largest area triangle that is contained within C_M? Guess: for any T, ...
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2answers
89 views

Smallest triangles that contain 2D convex regions with reflection symmetry

Given any 2D convex region $C$ with a mirror symmetry. Two pairs of questions: We need to find the smallest area (likewise, smallest perimeter) triangle that contains $C$. Is it sufficient to only ...
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1answer
196 views

What does the image of the integer lattice under a norm look like?

The question that I shall ask here has arisen in the context of Diophantine approximation. I find it rather interesting, and I have no idea how to answer it. Any help, advice, or suggestions for ...
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0answers
153 views

Generic shadows of convex bodies

If $K \subset \mathbb{R}^m$ is a convex body (i.e., a compact convex set) and $T \mathbin\colon \mathbb{R}^m \to \mathbb{R}^n$ is a linear map then $TK \subset \mathbb{R}^n$ is a convex body as well. ...
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0answers
26 views

Vertex enumeration for polytope with a sparse halfplane description?

Say I have a (bounded convex) polytope $P\subset\mathbb R^d$ with description $Ax\le b$, where $A$ is sparse in the sense that there are at most $k$ nonzero entries in each row or column, where $k$ is ...
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1answer
60 views

Convex-like properties of the polar parametrization of the boundary a convex body on the plane

Let $B$ be a compact convex set on the complex plane, containing zero in its interior. The boundary $\partial B$ of $B$ has the polar parametrization $\mathbf p:\mathbb R\to \partial B$ assigning to ...
3
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1answer
153 views

Quotient space of a locally uniformly rotund space

If $X$ is a uniformly rotund space , then for any closed subspace $M$ of $X$, $X/M$ is uniformly rotund. Does this hold for a locally uniformly rotund space? That is if $X$ is locally uniformly ...
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0answers
141 views

Minkowski sum, zonotopes, convex hull

For any set $P,Q$ in the Euclid space, define Minkowski sum '+' as follows: $P+Q=\{p+q|p\in P, q\in Q\}$. And define 'zonotope': a zonotope is the Minkowski sum of some (finite) segments (for example, ...
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1answer
137 views

How to define “interior” for the unit arc? [closed]

Let the unit arc be, $$\{x \in \mathbb{R}^2| x_1^2 + x_2^2 =1, x_1 \geq 0, x_2 \geq 0\}$$ There is something I found curious about the unit arc which is that, It has an empty interior viewed as a ...
1
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1answer
67 views

(Non-topological) interior of a convex set

Given a convex subset $X$ of a real vector space $V$, I'm interested in the set $$Y:=\{x\in X:\ \forall v\in V, \ \exists\epsilon>0 \text{ s.t. } x+\epsilon v \in Y \}.$$ My question is boring: ...
3
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1answer
68 views

On the area-perimeter ratio of a convex limited set

(Previously asked on MSE) Let $C\subset \mathbb{R}^2$ be a convex limited set. We define the average radius of $C$ as $$a_C=\frac{\int_{v\in C}d(v,C)dxdy}{A(C)}$$ Where $d(v,C)$ is the distance ...

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