# 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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### We have two convex sets $S,T$ such that $S\subseteq T$. Prove or disprove that the circumference of $S$ must be smaller than that of $T$. [duplicate]

This question originated from an exercise asking to compare the arclength of $y=x^2$ between $(0,0)$ and $(1,1)$ and $\frac{\pi}{2}$. The solution starts by constructing the circle with center $(0,1)$ ...
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### What is a convex combination of graphs?

For example in this paper, they refer to a "convex combination of trees" (pg. 2, first paragraph), and also, more generally, to "convex combination of graphs" (pg. 2, footnote). -&...
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### Two questions about Convex Sets [closed]

There are 2 questions that I can't make any idea while solving. Can you give an idea how I can solve it? For C convex, show that C is closed if and only if C $\cap$ L is closed, for any affine line L?...
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### Can a union of $k$ half spaces be partitioned into $O(k)$ disjoint convex polyhedrons?

Suppose $S=\bigcup_{i=1}^k S_i$, where $S_i=\{x\in \mathbb{R}^d|a_i^Tx\le b_i\}$ is a half space. Can we partition $S$ into $k$ disjoint convex polyhedrons? With the $k$ inequalities defining the $S_i$...
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### Range of z for convex constraint?

What would the range of z be if this were to be a convex constraint? $2x_1^2 + (2+z)x_2^2 - x_3^2 \leq 5$ I thought it could be approached by applying the two conditions of a convex hull (i.e, the ...
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### Name for a closed shape that has an interior point such that every line through the point meets the shape's boundary at exactly $k$ distinct points?

Consider a closed shape/set $S \in \mathbb{R}^2$ such that there exists an interior point $(x, y) \in S$ (need not be unique) such that any line passing through $(x, y)$ intersects $S$'s boundary at ...
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### Convex description of the positive orthant without the y-axis

I know that the following set is convex \begin{equation} X = \{(x,y) \in \mathbb{R}^2 : x>0 ,~ y\geq0 \}~ \cup~\{(0,0)\} \end{equation} i.e. the positive orthant without the y-axis but with the ...
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### Interior points in a convex set can be represented as convex combination of different points from the set

Can we assume that any interior point $z$ in a convex set $S\subseteq R^n$ be represented by $2$ points $x \in S$ and $y \in S$ such that $z = \lambda x +(1-\lambda)y$, where $x\neq y \neq z$ , and ...
1 vote
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### "Convexity" of a family of distributions

Let $\alpha\in(0,1)$. Consider a family of CDFs $\mathcal{X}$ that contains every CDF $X$ defined on $[0,\infty)$ with increasing hazard rate (IHR) which satisfies $$\mathbb{P}[x>y]=\alpha,$$ where ...
1 vote
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### Membership Oracle (MO) for $K+\epsilon B$ given MO for $K$ [closed]

Let $K$ be a convex compact set in $\mathbb{R}^d$. Suppose that a Membership Oracle $\mathcal{M}_{K}$ for the set $K$ is available; given a point $x\in \mathbb{R}^d$, the Oracle returns ''true'' if $x$...
1 vote
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### Request detailed explanation about Stephen Boyd cvxbook-solutions-manual exercise 2.8(a) expressing a set S in the form S = {x | Ax<=b, Fx = g}

cvxbook-solutions exercise page-5 exercise 2.8(a) 2.8(a) $S = \{y_1a_1 + y_2a_2 | − 1 ≤ y_1 ≤ 1, − 1 ≤ y_2 ≤ 1\}, \text{where }a_1, a_2 ∈ R^n$. The following is the solution mixed with my question. ...
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### Parametrize continuous deformation from one convex body to another

Suppose one has a continuous deformation, $Z(s)$, from an initial zonotope $Z(0)=Z_0$ to a final one $Z(1)=Z_f$, such that $Z(s)\subset Z(s^\prime)$ whenever $s<s^\prime$. I need to define ...
I have given the following cone: $P=\lbrace x | Ax \geq 0\rbrace$ where $A=\begin{pmatrix} -1& 1 & 0 & 0 & 0&0\\ 0& -1 & 1 & 0 & -1 & 0\\ 0 & 0 & -1 &... 0 votes 0 answers 26 views ### Best way to describe distance from surface point to projected interior point Is there a better way to describe the following idea in words? I've been really struggling. My first thought is to call this "skewness" but I don't know where I got that from and it might ... 0 votes 0 answers 25 views ### Show that the$l_1$ball is a convex set based on the convexity of polytope$P$is defined as the set of points$x \in \mathbb{R}^d$satisfying the following constraints: for an integer$m > 0$, for$m$vectors$a_1, ..., a_m \in \mathbb{R}^d$and$m$values$b_i \in \...
According to Royden: Mazur's Theorem: Let $K$ be a nonempty convex subset of a normed linear space $X$; $K$ is strongly closed if and only if it is weakly closed. According to Wikipedia: Mazur's ...