# Questions tagged [convex-geometry]

Use this tag when posting questions related to the concept of convexity and geometry. For example, for convex polygons.

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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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### 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 ...
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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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### 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 ...
1 vote
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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)&...
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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 ...
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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 ...
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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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### 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 ...
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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 ...
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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 ...
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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 ...
1 vote
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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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### 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{...
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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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### 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 ...
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 $$\... 2 votes 0 answers 57 views ### Is there a convex polyhedron all of whose 2D polygonal cross sections are asymmetrical? Define a nondegenerate convex n-polytope as the convex hull of a finite number of points in Euclidean n-space, such that its interior is nonempty. It seems true that every nondegenerate convex ... -2 votes 1 answer 40 views ### Convex cones in \mathbb{R}^n - Orthants I am trying to understand the notion of convex cones. So, here are my questions. I can understand that the non-negative orthant, \mathbb{R^n_+}, defined as \left\{ (x_1, \ldots, x_n) \in \mathbb{R}^... 0 votes 0 answers 22 views ### How to show after small change the intersection between two convex shapes still belongs to an open set Assume two convex shapes A and B in \mathbb{R}^3 (both A and B are compact sets in \mathbb{R}^3), and denote their intersection as C = A \cap B. Let C belong to an open set D, i.e., ... 0 votes 0 answers 14 views ### (\prod_1^n (1+c_i t))^\frac{1}{n} is concave in t I would like to show that f(t) = (\prod_{i=1}^n (1+c_i t))^\frac{1}{n} is concave on [0,1]. I think it is related to AM-GM inequality. f(t) is GM, while corresponding AM is linear and is always ... 0 votes 0 answers 27 views ### Measure of Cylinder Intersecting Sphere in \mathbb{R}^n Please check my work. Let \mu be the uniform measure on \mathbb{S}^n, the unit sphere with radius 1, and$$A=\{x\in\mathbb{S}^n:x_1^2+x_2^2\le \sin^2\alpha\}$$where \alpha is constant. Claim ... 0 votes 0 answers 20 views ### Is it ture that any zonoid (not a trivial zonotope), must be a affine image of a unit ball (l2)? I am trying to find some literature discussing zonoid in 3-dimensional space. And a seemly simple question that I am considered specifically is what are those non-trivial (not being a zonohedron) ... 0 votes 0 answers 20 views ### Given the support function of a convex set C\subseteq \mathbb{R}^{2n}, compute \sup \left \{c'y: (x,y) \in C\right\} as a function of x Suppose C \subseteq \mathbb{R}^{2n} is a closed, bounded, convex set, with support function h: \mathbb{R}^{2n} \rightarrow \mathbb{R}, defined as$$h(c_1, c_2) := \sup \{c_1'x + c_2'y : (x,y) \in ...
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 ...