Questions tagged [convex-geometry]
Use this tag when posting questions related to the concept of convexity and geometry. For example, for convex polygons.
1,347
questions
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If $D$ isn't a convex billiard table, then the billiard map $T$ is not continuous
I am currently reading chapter 3 of "Geometry and Billiards" by Serge Tabachnikov, and I have some doubts about the need of using convex sets.
So, here's how the billiard map is defined:
To ...
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20
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What is the smallest ball where a convex body of given diameter belongs to?
Let the diameter of the convex body $K$ be $c.$ It's easy to see that if the convex body is symmetric, then the convex body belongs to a ball $B(y,c/2).$ This bound is sharp in every dimension. But ...
4
votes
1
answer
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Convexity of a connected, compact, and locally convex set in $\mathbb{R}^n$
Is a compact, connected, and "locally convex" set in $\mathbb{R}^n$ convex?
Here I mean a space $A$ locally convex as: For any point $x\in A$, there exists a neighborhood $U$ of $x$ s.t. $U$...
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1
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Given a convex polygon with integer coordinates vertices, how can you transform it so you can always find a point inside it with integer coordinates?
Given a convex polygon, with all its vertices having integer coordinates, you are allowed to perform one transformation T to all of its vertices (which are points), such that:
T transforms a point in ...
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What is the connection between the gradient acting on the support function in Euclidean space and on a sphere?
The support function defined on the unit sphere can be locally represented as a support function defined in the entire space. This can be achieved by using a mapping
\begin{align}
(x_1,\cdots,x_n)&...
- 23
0
votes
1
answer
40
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Estimating Vectors from the Convex Hull
Let us consider a set of $N$ vectors, $\mathbf{h}_1, \mathbf{h}_2, \cdots, \mathbf{h}_N$, such that $\mathbf{h}_i \in \mathbb{R}^{M}$, $\forall i$, with $M < N$.
Let us also consider the space $\...
- 25
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6
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Convexity of minkowski space when two triangles collide
I am working on a Python program to show the collision of two triangles by using Minkowski difference. I am subtracting each point from one triangle from every other point on the other triangle. The ...
2
votes
2
answers
57
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Dimension of a polytope cut with a hyperplane
Let $P \subseteq \mathbb{R}^d$ be a convex polytope and let $H \subseteq \mathbb{R}^d$ be a hyperplane with two sides $H^+$ and $H^-$. Let $V$ be the vertex set of $P$. Suppose $V = A \cup B$ where $A$...
1
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1
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If $K$ is a convex body and $\xi\in S^{n-1}$, is $F(x) = \left(\int_{x}^\infty {vol_{n-1}(K \cap (\xi^\perp + t\xi))dt}\right)^{\frac{1}{n}}$ concave?
I was trying to figure out whether this function $$F(x) = \Big(\int_{x}^\infty {vol_{n-1}(K \cap (\xi^\perp + t\xi))dt}\Big)^{\frac{1}{n}}$$ was concave on its support for any convex body $K$ and any ...
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For a proper cone $K$, $x \in \operatorname{int}(K), y \in K$, implies $x+y \in \operatorname{int}(K)$?
Let $K$ be a proper cone in $\mathbb{R}^{n}$, which is a cone that is convex, closed, pointed, and has a nonempty interior. If $x \in \operatorname{int}(K)$, and $y \in K$, must $x+y \in \operatorname{...
- 1,076
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Computing the smallest area for symmetric convex body around O in $\mathbb R^2$ that contains no two linearly independent unimodular lattices?
Let $\mathcal L$ be the space of unimodular 2-lattices in $\mathbb R^2$ (namely with covolume $1$). I believe there is a fact about primitive vectors (those who are not nontrivial integer multiple of ...
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Inequality for strictly increasing convex function [closed]
Suppose I have a strictly increasing convex function $f:[0,\infty)\mapsto [0,\infty)$. How do I show that
$$f^{-1}(x^2) \leq C_f f^{-1}(x)$$ for any $x\geq 0$, and $C_f$ is a constant that is ...
- 353
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Projections of polyhedra are polyhedra?
Consider a (convex, compact, oriented) k-polyhedron $P\subset \mathbb{R}^N$, and $U\subset \mathbb{R}^N$ any $n$-dimensional linear subspace. Let $\pi_U:\mathbb{R}^n\to U$ denote the orthogonal ...
- 1,662
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A domain is convex if and only if it is contained in its tangent half spaces.
Let $D$ be an open domain in $\mathbb{R}^{2}$ whose boundary $\partial D$ is a 1-dimensional C${}^{\infty}$-submanifold of $\mathbb{R}^{2}$. Given a boundary point $x$ in $\partial D$, let $\nu(x)$ ...
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1
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Steiner symmetrization preserves compactness？
I have read " Convexity, H.G.Eggleston, 1958'' and in page 91, theorem 43, it proves that a closed, bounded and convex set $\mathcal{X}$ is still closed bounded (and convex) under Steiner ...
- 51
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Optimization Method with Circumference Level Curves and Affine Constraints
Hello fellow mathematicians,
I am a computer engineering PhD student specializing in control engineering, seeking your insights on an optimization method I've been working on. My approach revolves ...
3
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Zeroing out one component of a vector that is in a convex set
Let $C$ be a closed convex set in $\mathbb{R}^n$ that contains zero.
Let $x \in C$ be a vector with $n$ components. Suppose we zero out one of its components. Without loss of generality, let that ...
- 113
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On the convexity of a certain set in $\mathbb{R}^n$
Let $\mathcal{P}_{i,\rho_i}$ be a polytope in $\mathbb{R}^n$ defined as follows
$$ \begin{cases}
A_{i,1}^T\cdot x + b_{i,1} \leq \rho_i \newline
\vdots \\
A_{i,m} \cdot x + b_{i,m} \leq \rho_i
\end{...
- 1,237
1
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1
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Determining if an over constrained system of linear inequalities has a solution?
I have a system of inequalities of the form:
$$ X \cdot v_i \leq 0 $$
For some number of vectors $v_i$.
I need to determine if there exists at least one $X$ that can satisfy all of them at once (I don'...
- 3,088
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1
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Prove that for any small enough translation of a simplex either the new simplex has a vertex of the old simplex or viceversa
Consider a $n$ dimensional simplex. Prove (or disprove) that a small enough translation of it in any direction produces a simplex which either has a vertex of the old simplex, either the new simplex ...
- 1,237
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31
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Convex sets and extreme points
I am learning about convex sets and extreme points from a course on linear programming. I came across a theorem that states that every closed convex set has an extreme point if and only if it does not ...
- 235
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37
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Convexity of spherical curves
Let $\gamma$ be a smooth curve in $\mathbb{R}^{3}$. We say that $\gamma$ is convex if it lies on the boundary of its convex hull. Recall that the convex hull of $\gamma$ is the set of all convex ...
user242708
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answers
69
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Convex hull of convex space curve
Let $\gamma$ be a smooth closed curve $\mathbb{R}^{3}$. We say that $\gamma$ is convex if it lies on the boundary of its convex hull, which we denote by $\mathrm{conv}(\gamma)$.
I know that the convex ...
user242708
0
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35
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$k$-dimensional faces of an $n$-dimensional cube
I want to find a formula that shows the number of $k$-dimensional faces of an $n$-dimensional cube. By internet I found that this formula has a generating function, $(x+2)^n$, where the formula is the ...
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1
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56
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Inequality regarding strong convex domain
Suppose $\Omega\subset\subset \mathbb{R}^n$, be a bounded, strongly convex domain (with $0\in\Omega$). Is the following inequality true?
$$
C_2\left|x-y\right|^2\leq\left|\frac{x}{|x|}-\frac{y}{|y|}\...
- 43
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answers
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Equivalent characterizations of inscribed rectangle in convex region
Note 1 in this MO question seems to suggest the maximal area inscribed rectangle in a planar convex region is also the maximal perimeter rectangle in convex shape. I am not entirely convinced this is ...
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1
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How many times do two circles intersect?
Suppose you have two compact convex sets $A,B \subset \mathbb R^2$. Their interiors intersect, but neither is contained in the other. Do the boundaries intersect in at least two distinct points?
...
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1
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Convexity of the lognormal skewness-kurtosis curve
The skewness-kurtosis curve of the family of lognormal random variables can be parametrized as
\begin{eqnarray*}
x(t) &=& (t+2)\sqrt{t-1} \newline
y(t) &=& t^4+2t^3+3t^2-3
\end{...
- 100
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How to solve the primal problem via Lagrangian directly?
From section 5.5.5 of Convex Optimization by Stephen Boyd and Lieven Vandenberghe https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf, consider the primal problem
$$\begin{cases}
\min:\ f_0(x)\\
\...
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Does the origin have a strictly convex bounded neighborhood?
Given any finite-dimensional normed space, the topology is equivalent to that generated by the Euclidean norm. So the Euclidean open ball is norm closed. The fact that is is strictly convex is ...
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Cutting segments from convex objects
Let $X$ be some convex object in the plane. For every pair of points $A,B$ that are not in $X$, and the distance between them is at most $1$, we remove from $X$ its intersection with the segment $AB$. ...
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Clarification on a step of the proof of the 4 vertex theorem
We are given a simple plane closed convex curve. Its curvature, being continuous on the entire curve, which is compact, achieves its maximum and minimum there, say at points $p$ and $q$. The author ...
- 1,261
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Surface area of L2 ball + L1 ball
Consider the L1-ball $A$ and the L2-ball $B$ in $d$-dimensions.
Consider the Minkowski sum $A+B$ of the two corresponding sets of points.
Is there a known expression for the Vol$(A+B)$ or the surface ...
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Convex hull of a set of $2^{n-2}$ points without a convex subset of $n$ points
I was recently looking at the happy ending problem, and on the linked Wikipedia page, there is the following conjecture (the Erdos-Szekeres conjecture), which states the following:
The smallest ...
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1
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How many probability vectors do I need to convexly generate any probability vector?
Fix a positive integer $n$, and let $B = \{p_1, \dots, p_{k}\}$ be a set of $k$ mutually linearly independent vectors in $\mathbb{R}^{2^n}$, where each $p_i \in B$ is also a probability vector in the ...
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How do I prove that hyperbolic triangles are convex?
I was introduced to both the Poincaré disc model and the upper half-plane model and need to prove the statement in the title. Apparently you can do this by showing that sectors in the disc model with ...
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The lower bound of the ratio of the diameter of a convex polygon to its perimeter
Let's say I have a convex polygon $P$. $D$ is its diameter (the distance between its two farthest-apart vertices) and $\varphi$ is its perimeter. I believe the following to be true:
$$D/\varphi \ge 1/\...
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Degenerate convex function that is not finite on the closure of some convex set on which the function is finite.
Let $E$ be a locally convex TVS, $C$ a convex subset of $E$ and $g:E\to]-\infty,\infty]$ a convex function. I was trying to prove the following
If $g$ is finite on $C$, then $g$ is finite on $\...
- 5,385
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2
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Find the conditions which guarantee a polyhedron is bounded (i.e., a polytope)?
Note, a polyhedron is the intersection of finitely many half spaces in $\mathbb{R}^n$ and a polytope is a bounded polyhedron.
Preliminary: Consider a hyperplane $C_1x_1 + C_2x_2 + \dots + C_nx_n + D = ...
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Find the constraints which guarantee that 6 planes in 3D space form a convex box containing the origin?
I ever asked a question to find the constraints which ensure 4 lines in 2D space form a convex quadrilateral; see link and it has been solved by @YNK author perfectly.
Now I hope to extend it and came ...
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39
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affine combinations of Bernstein basis polynomials that are nonnegative and sum to 1
The $i^\text{th}$ degree-$n$ Bernstein basis polynomial is defined as
$$
\begin{equation}
b_i^n(x) = \binom{n}{i} (1-x)^{n-i} x^i.
\end{equation}
$$
The Bernstein basis polynomials have many ...
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Relation between volume of a convex polytope and its width
Let $P=\{x\in \mathbb{R}^n\mid a_i^\top x\leq b_i,\, i=1,\ldots,m\}$ be a bounded convex polytope, $\|a_i\|_2=1,\, i=1,\ldots,m$. Let us define the width of $P$ in the direction $a_i$ as
$$
\...
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Is there a convex polyhedron all of whose 2D polygonal cross sections are asymmetrical?
Define a nondegenerate convex n-polytope as the convex hull of a finite number of points in Euclidean n-space, such that its interior is nonempty.
It seems true that every nondegenerate convex ...
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Convex cones in $\mathbb{R}^n$ - Orthants
I am trying to understand the notion of convex cones. So, here are my questions.
I can understand that the non-negative orthant, $\mathbb{R^n_+}$, defined as $\left\{ (x_1, \ldots, x_n) \in \mathbb{R}^...
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votes
0
answers
22
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How to show after small change the intersection between two convex shapes still belongs to an open set
Assume two convex shapes $A$ and $B$ in $\mathbb{R}^3$ (both $A$ and $B$ are compact sets in $\mathbb{R}^3$), and denote their intersection as $C = A \cap B$. Let $C$ belong to an open set $D$, i.e., $...
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0
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$(\prod_1^n (1+c_i t))^\frac{1}{n}$ is concave in t
I would like to show that $f(t) = (\prod_{i=1}^n (1+c_i t))^\frac{1}{n}$ is concave on $[0,1]$.
I think it is related to AM-GM inequality. $f(t)$ is GM, while corresponding AM is linear and is always ...
- 485
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0
answers
27
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Measure of Cylinder Intersecting Sphere in $\mathbb{R}^n$
Please check my work.
Let $\mu$ be the uniform measure on $\mathbb{S}^n$, the unit sphere with radius $1$, and $$A=\{x\in\mathbb{S}^n:x_1^2+x_2^2\le \sin^2\alpha\}$$
where $\alpha$ is constant.
Claim $...
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Is it ture that any zonoid (not a trivial zonotope), must be a affine image of a unit ball (l2)?
I am trying to find some literature discussing zonoid in 3-dimensional space. And a seemly simple question that I am considered specifically is what are those non-trivial (not being a zonohedron) ...
- 11
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0
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20
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Given the support function of a convex set $C\subseteq \mathbb{R}^{2n}$, compute $\sup \left \{c'y: (x,y) \in C\right\}$ as a function of $x$
Suppose $C \subseteq \mathbb{R}^{2n}$ is a closed, bounded, convex set, with support function $h: \mathbb{R}^{2n} \rightarrow \mathbb{R}$, defined as
$$h(c_1, c_2) := \sup \{c_1'x + c_2'y : (x,y) \in ...
- 1,584
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1
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A disk without a point on its boundary is a convex set?
I trying to show that any homeomorphism $h:D^2\to D^2$, where $D^2 =\left\{x\in \mathbb{R}^2: |x|\leq 1\right\}$ takes $S^1$ to $S^1$ and ${D^2}^\circ$ to ${D^2}^\circ$(interior of a disk).
Supposed ...
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