1
vote
1answer
20 views

Show that $\mathcal{C}(A)$ is the smallest convex of $E$ containing $A$.

Let $E$ be a $\mathbb{R}$-vector space, and $A$ a nonempty subset of $E$. Show that $$\mathcal{C}(A) = \biggl\{\sum \limits_{k=1}^n \lambda_kx_k \biggm| n \in \mathbb{N}^*, (x_1,\dots,x_n) \in ...
0
votes
0answers
20 views

Separating convex sets in a tvs $X$.

I got doubt with the proof of this theorem. Let $X$ be a tvs, $A,B \subset X$ with $A$ an open convex set and $B$ convex such that $A \cap B = \emptyset$. Then there exists $f \in X^*$ (where ...
1
vote
0answers
17 views

Checking convexity by looking at 2-dimensional cross-sections

If I have a closed set of n-dimensional points and I want to know if it's convex just by examining some set of 2-dimensional cross-sections (and checking each cross-section for convexity), how small ...
1
vote
2answers
389 views

Prove Convex Hull of Minkowski sum

I want to prove that the following holds, where the $+$ means Minkowski sum: $$ conv(A+B)=conv(A)+conv(B) $$ Let the convex hull of $A+B$ be $$ conv(A+B)=\sum_{j,k}\lambda_j\mu_k(a_j+b_k) $$ I ...
1
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0answers
31 views

What is first edge position in the Minkowski sum of two convex polygons in the plane?

I am trying to understand the informal algorithm of the Minkowski sum of two convex polygons in the plane as described here: ...
0
votes
1answer
22 views

Conjugate vectors

What are conjugate vectors? Can I have an example of it? [ This question is in respect to finding the roots of equations with conjugate direction methods]
0
votes
2answers
164 views

Prove that the following two definitions of the convex hull are equivalent.

I was wondering if a topology expert could help me solve this proof, as I have no idea but want to understand these concepts. This is not for homework. Let X be a point set, not necessarily finite, ...
0
votes
1answer
40 views

Terminology for a weaked vector space

Let $S$ be a semigroup on which acts $\mathbb{R}_{\geq0}$. Does this structure has a name? For example $S$ can be the set of convex bodies in $\mathbb{R}^n$ with the Minkowsky sum.
0
votes
1answer
56 views

Convex Function of Two Vectors

Let $f:\mathbb{R}^M\times\mathbb{R}^N\rightarrow\mathbb{R}$ be a mapping such that for $\mathbf{Y}\in\mathbb{R}^N$ constant, $f(\mathbf{X}, \mathbf{Y})$ is a convex function of $\mathbf{X}$ and for ...
2
votes
2answers
106 views

Convex cone question.

Hoi, let $V$ be finite dimensional real vector space with inner product $\left\langle . \right\rangle$ and let $\Gamma \neq \{0\}$ be a closed convex cone. Let $$\Gamma_0^{\perp}:=\{v\in ...
0
votes
1answer
67 views

Pointed Convex cone: one-to-one correspondence extreme rays - extreme points

Hoi, let $V$ be a finite dimensional real vector space with inner product $\left\langle .\right\rangle$. Let $\Gamma\subset V$ and $\Gamma \neq \left\{0\right\}$ a pointed convex cone. (Pointed means ...
2
votes
1answer
204 views

Projection operator property

Let $\pi_M(a)$(projection operator) be the closest point of $M$ from the point $a$ . How one can prove if $M$ is convex set of $\mathbb R^n$ then projection operator has this property? ...
2
votes
1answer
63 views

Definition of direct products of two cones or of two convex subsets?

When reading a comment after this reply, I was wondering what the definitions of direct product of two cones? More generally, what is the direct product of two convex subsets? This case is what I ...
1
vote
2answers
129 views

Convexity and minimum of a vector function

Prove that the function $f:\mathbb{R}^n\to \mathbb{R}$ given by $f(x)=x^T \cdot x$ is strictly convex. Use this result to find the absolute minimum by equating the derivative to zero. I am not sure ...
1
vote
2answers
267 views

Infimum/Supremum of an intersection of subsets of a vector space.

Consider a collection of subsets $A_i$ of an ordnered vector space or field. I am trying to find out, under what (minimal) conditions the following holds. $\inf \bigcap_{i\in I}A_i \le \sup_{i\in ...
1
vote
5answers
194 views

Is closure of convex subset of $X$ is again a convex subset of $X$?

Today I was going through my text book exercise and got hold of the following question. Let $X$ be a normed linear space with norm $\lVert\cdot\rVert$ and $A$ is a non empty convex subset of $X$ ...
1
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1answer
92 views

Question about piecewise linear paths

I'm having trouble with a concept about piecewise linear paths. I've been looking on the net but I haven't found a definition of it. Consider a finite family of hyperplanes in a finite-dimension real ...
4
votes
1answer
69 views

A question about pyramids (polytopes)

Let's use the following definition of a face: A nonempty convex subset $F$ of a convex set $C$ is called a face of $C$ if $\alpha x + (1-\alpha) y \in F$ with $x, y \in C$ and $0 < \alpha < 1$ ...
1
vote
2answers
437 views

Two definitions of a face of a convex set: are they equivalent?

I am used to the following definition of a (proper) face of a polytope: A nonempty convex subset $F$ of a polytope $C$ is called a face of $C$ if $\alpha x + (1-\alpha) y \in F$ with $x, y \in C$ and ...
1
vote
1answer
241 views

convexity of inner product of elementwise powers

For $x \in \mathbb{R}^n$ and $A,B \in \mathbb{R}^{m \times n}$, $f(x) = ((Ax)^{2})^T((Bx)^2)$ where $^2$ denotes the power of 2, element-by-element of vector Ax or Bx. (I wasn't sure how to notate ...