Tagged Questions
0
votes
0answers
34 views
The explicit set-description for the intersection of a convex and affine hull
When given two finite sets of points in $\mathbb{R}^n$, how does one get an explicit description of their convex (resp. affine) hull?
E. g. Suppose
$
A = \{e_1, e_2, e_3, e_4\},\;
B = ...
2
votes
4answers
85 views
Sufficient condition for a function to be affine
If for a function $f:\mathbb{R}\to\mathbb{R}$, I can prove for any real $x,y$, that $f(\frac{x+y}{2})=\frac{f(x)}{2}+\frac{f(y)}{2}$, can I say for sure that it is affine, as in of the form ...
1
vote
0answers
64 views
Convexity of affine function.
Can someone help me with a proof that affine function preserves convexity?
Given that $f$ is convex, $A$ is in $\mathbb{R}^{M\times N}$ and $b$ is in $\mathbb{R}^m$ then show that $g(x) = f(Ax+b)$ is ...
0
votes
1answer
73 views
affine set convex set
How to show the following:
Let $C$ be a convex subset of $\mathbb R^d$. Then $\operatorname{int} C \neq \emptyset$ if and only if
$\operatorname{aff} C = \mathbb R^d$ where $\operatorname{aff} C$ is ...
6
votes
3answers
305 views
Why is the affine hull of the unit circle R^2?
In Boyd's "Convex Optimization" it defines the affine hull of a subset $C$ of $R^n$ as
$$\text{aff} C = \left\{\theta_1 x_1 + \ldots +\theta_k x_k \mid x_1, \ldots x_k \in C, \theta_1 + \ldots ...
0
votes
1answer
43 views
Confusion regarding convex and affine set
I am a bit confused regarding convex and affine set. When they mention set, does it mean the set consisting of all the points belonging to the line or shape respectively?
0
votes
0answers
73 views
distance of an affine subspace to a polytope
I wonder how to prove the following statement.
Let $V$ be a $d$-dimensional normed space with $d \geq 3$, let $P \subset V$ be a $(d-2)$-dimensional polytope. Then there is an $\epsilon > 0$ such ...
3
votes
1answer
154 views
Effect the zero vector has on the dimension of affine hulls and linear hulls
I am currently working through "An Introduction to Convex Polytopes" by Arne Brondsted and there is a question in the exercises that I would like a hint, or a nudge in the right direction, please no ...
3
votes
0answers
247 views
Is every convex-linear map an affine map?
Let's say that a map $f: V \rightarrow W$ between finite-dimensional real vector spaces is convex-linear if $f(\lambda x + (1-\lambda)y) = \lambda f(x) + (1-\lambda)f(y)$ for all $\lambda \in [0,1]$.
...
