Questions tagged [convexity]
For questions involving the concept of convexity
625
questions
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Are convex functions on manifolds the same as $c$-convex functions, where $c(x,y)=d(x,y)^2/2$?
I am reading the following book on optimal transport. While reading I came across the following definition of $c-$convexity.
Given $X$ and $Y$ metric spaces, $c: X \times Y \rightarrow \mathbb{R}$, ...
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0
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51
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"More" cyclical monotonicity
Let $X$ and $Y$ be some finite sets. For a given function $f:X\times Y\rightarrow \mathbb{R}$, we say a set $S\subset X\times Y$ is $f$-cyclically monotone if for any sequence $(x_1,y_1),...,(x_n,y_n)\...
2
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2
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183
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Elementary convexity example
I'm trying to check that certain examples of Young functions in the harmonic analysis literature are actually Young functions, and in doing so need to prove the following convexity-like inequality for ...
0
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1
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121
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Strict inclusion for recession cone of closure of a convex set
Let $C$ be a nonempty closed convex subset of $\mathbb{R}^n$. The recession cone of $C$ is given by
$$R_C=\left\lbrace d\in\mathbb{R}^n:x+td\in C, \forall t>0, \forall x\in C\right\rbrace.$$
It is ...
1
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0
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168
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The image of zero-measure set under normal mapping is Lebesgue measurable
Let $u$ be a convex function defined on a bounded open set $\Omega$ in $\mathbb{R}^n$. Then $u$ is twice differentiable a.e. Let $E_u$ be the set on which $u$ is not twice differentiable. Then $E_u$ ...
6
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2
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300
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For most directions does the supporting hyperplane meeting a bounded convex set meet it in one point?
Let $C\subseteq \mathbb R^n$ be non-empty, convex and compact. For $v\in S^{n-1}$, let $H_v$ be the supporting hyperplane in the direction of $v$ (i.e., $H_v$ is the boundary of the smallest closed ...
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3
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Convergence of convex functions
I can prove the following result.
Theorem 1. Let $f_n:\mathbb{R}^n\to \mathbb{R}$ be a sequence of convex functions
that converges almost everywhere to a function $f:\mathbb{R}^n\to\mathbb{R}$.
Then ...
7
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0
answers
109
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What is the closed cone generated by constant and coordinate functions and closed under taking $f\mapsto\max(f,0)$?
Let $C$ be the smallest closed convex cone of functions from $\mathbb{R}^n$ to $\mathbb{R}$ that contains all constant functions, all coordinate functions, and such that $\max(f,0)\in C$ whenever $f\...
0
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0
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161
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Lipschitz map on positive definite cone of $n$-by-$n$ matrices
A function matrix $f : X \to \mathbb R$ is a convex Lipschitz continuous matrix function with Lipschitz constant $\mathrm L$ with respect to a fixed given norm $\|\cdot\|$, i.e., $|f(A)-f(B)| \leq \...
2
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Intersection of the simplex with a linear subspace of codimension $2$
The sets are defined in $\mathbb{R}_+^n$ $(n\geq 1)$. The relative interior of a convex set $C$ is denoted $\mathring C$.
Let $S$ be the $n$-simplex:
$$S=\left\{x\in\mathbb{R}_+^n,\,\sum_{i=1}^n x_i=1\...
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1
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111
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Can we use the solution to two optimisation problems to solve a third, bigger, one?
Background
Say we have an optimization problem $$\min_x f(x) = g(x) + h(x)$$
where $g$ is differentiable and convex, and $h$ are convex but not necessarily differentiable. If $g$ is the mean squared ...
1
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1
answer
148
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A question on convexity and conjugate points
Let $(M,g)$ be a compact smooth simply connected Riemannian manifold with a smooth boundary. Assume also that $(M,g)$ does not have any conjugate pairs of points. Let $\Gamma \subset \partial M$ be a ...
2
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1
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322
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Concavity/convexity of distance-to-boundary function
For $\Omega$ a bounded open set of $\mathbf{R}^d$, denote $\mathrm{d}_\Omega:x\mapsto \mathrm{d}(x,\partial\Omega)$ the distance-to-boundary function.
If $\Omega$ is convex, a short argument recalled ...
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2
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218
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Decreasing magnitude of spherical centroid
Let $\sigma$ be the uniform measure on $\mathbb{S}^{d-1}\subset \mathbb{R}^d$. For any region $R\subset \mathbb{S}^{d-1}$, let $X_R$ be a random variable which is uniformly distributed across $R$. We ...
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99
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Literature request: proving or disproving convexity of the optimal value function of semidefinite program (SDP) or convex optimization in general
Suppose I have a function $f:\mathbb{R}\rightarrow \mathbb{R}$ defined as the following parametric optimization problem:
$$f(p) = \inf_xf_0(x) \quad \text{subject to } \quad G(x,p)\leq 0,$$
where ...
7
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2
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438
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Proving the set $\left\lbrace \frac{(x + y)^2}{\sqrt{y}} \leq x - y + 5, y > 0 \right\rbrace$ is convex
I have recently picked up a course on Convex Analysis in my spare time, but feel I'm not quite up to speed with the 'tricks' for proving a set is convex.
I have managed to prove this by moving all ...
2
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1
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328
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Are there "pathological convex sets" over ultravalued fields of char 2?
In their book Topological Vector Spaces (2nd ed.) Lawrence Narici and Edward Beckenstein generalise convex sets for TVS over ultravalued field $K$ as $K$-convex sets. The definition goes as following:...
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173
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Lipschitz aspect of a projection on the boundary of a convex
Let $C$ be a closed convex set of $\mathbb{R}^n$ $(n\geq 1)$, with asymptotic cone $C^{as}$ having for interior $\text{Int}\big(C^{as}\big)$. Let $u\in\mathbb{R}^n\setminus\{0\}$ such that
\begin{...
4
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2
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363
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Is this projection on the boundary of a convex Lipschitz?
Let $C$ be a closed convex set of $\mathbb{R}^n$ $(n\geq 1)$, and $u\in\mathbb{R}^n\setminus\{0\}$ such that $u$ does not belong to the asymptotic cone of $C$ and is nowhere tangent to $\partial C$. ...
5
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0
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101
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Semilinear elliptic equation
Assume $u$ is a smooth solution for
$$
\Delta u + f(u)=0\qquad \hbox{in}\quad \Omega
$$
and $\Omega$ is a smooth convex domain in $\mathbb{R}^n$.
Is there a conjecture which are the weakest conditions ...
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1
answer
237
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Intersecting points of increasing convex functions
Can two increasing and differentiable convex functions agree exactly on a countable set of cardinality greater than two?
3
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219
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Hausdorff dimension of the non-differentiability set a convex function
Let $X \subset \mathbb R^d$ be open, $f : X \to \mathbb R$ and
$$
E := \{x \in X : f \text{ is not Fréchet differentiable at }x\}.
$$
Then we have the following result which is
Theorem: If $X= \...
1
vote
1
answer
88
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Volume ratio of polytopes with few vertices
The volume ratio of a convex body $K\subset \mathbb{R}^{n}$ is $v_r(K) = \inf_{\mathcal{E}\subset K} \left(\frac{Vol(K)}{Vol(\mathcal{E})}\right)^{1/n}$ where the infimum run over ellipsoids included ...
2
votes
1
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158
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Gradient of a convex function on $\mathbb{R}^d$, maximum on hypercubes bounded by values in corners?
Let $f : \mathbb{R}^d \rightarrow \mathbb{R}$ be infinitely often continuously differentiable and convex.
For $d = 1$, we know that for any interval $[a, b]$, it holds for $x, y \in [a, b]$ that
$$
(f'...
6
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1
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349
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Convex solutions of the Poisson equation
Let $D$ be a planar, bounded, convex open domain. Given a positive function $f:D\to(0,+\infty)$, let us consider the Poisson equation
$$\Delta u=f\quad\hbox{in }D.$$
Not specifying any boundary ...
1
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1
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105
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Measurable sets of $\mathbb R^n$ forming unique absolutely continuous convex combinations?
If we consider a finite set $A\subset\mathbb R^n$, uniqueness of the convex decomposition of points in $A$ is equivalent to the absence of $\mu\neq0$ signed measure supported on $A$ such that $\mu(\...
2
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1
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275
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Convex series and closed convex hulls in normed spaces
Let $(X, \lVert \cdot \rVert)$ be a normed space over $\mathbb{R}$ and $A = \{ a_1,a_2 \ldots \} \subseteq X$ be a closed bounded set.
Let $\overline{\mathrm{co}}(A)$ denote the closed convex hull of ...
7
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2
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530
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A counterexample to: $\frac{1-f(x)^2}{1-x^2}\le f'(x)$ — revisited
Can we find a counterexample to the following assertion?
Assume that $f:[-1,1]\to [-1,1]$ an odd function of class $C^3$, and assume thaht $f$ is a concave increasing diffeomorphism of $[0,1]$ onto ...
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108
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Graduate level convexity - Intersection of an r-polytope with a hyperplane is an r-1 polytope
I am trying to follow Roger Webster's Convexity 's proof of Euler's celebrated result on the relationship between the number of faces of a polytope. An image of the proof is here.
In the course of the ...
3
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54
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Injectivity of a convex combination of squares $A^*A$ in $\ell^\infty$
Consider two operators $A,B: \ell^p \to \ell^p$ ([defined and]bounded for all $p \in [1,\infty]$) as well as their adjoints $A^*,B^*:\ell^p \to \ell^p$. Assume $A^*A$ and $B^*B$ have trivial kernel ...
4
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260
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A convexity question
Let $Q=[0,1]\times[0,1]$ and let $a$ be a positive smooth function on $Q$. Does there exist a smooth positive function $u$ On $Q$ such that there holds
$$ \frac{\partial^2}{\partial x_1^2}u <0 $$
...
0
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0
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96
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Sharp, salient and opposite cones
I have been reading about star shaped sets and support cones from this article.
Can anyone please help me with examples the difference between a sharp and dull cone.
How come a salient cone has a ...
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0
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166
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Regarding definition of convex cone and apex
I have been reading about star shaped sets and support cones from this article. I am wondering about the definition of the cone as to why is it defined this way? Why is it $C-a$? I am familiar with ...
3
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1
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616
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Is a convex, lower semicontinuous function that is bounded from below, actually continuous?
While thinking about convex functions, I managed to put together the following proof which I find a bit too good to be true. $X$ is a topological vector space that is also a Baire space.
Lemma: Let $f ...
4
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2
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How to prove $ \sum_{k=1}^{n}f(a_k)\leq nf\left(\frac{b}{n}\right) $ for sufficiently large $ n $ here?
Let $ 0<a<b $, $ f\in C^1\left([0,b]\right)$. Assume that $ f $ is concave on $ [0,a] $ and convex on $ [a,b] $ with $ f'(0)>f'(b) $. Please prove that there exist $ n_0\in\mathbb{N} $ which ...
3
votes
1
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231
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$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^1 f(\sin(1/x)) dx \times \int_0^1 g(\cos(1/x))dx? $
I have noticed experimentally that the following question has a positive answer.
Is it true that for all even and convex functions $f$, $g$:
$$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^...
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For $f$ geodesically convex with $L$-Lipschitz-gradient on hyperbolic space, is $f(x)-f(x^*)\leq(\mathrm{const}) \cdot L r$ for all $x \in B(x^*, r)$?
$\DeclareMathOperator\dist{dist}$Setting: Let $M$ be a hyperbolic space of sectional curvature $-1$, and let $f \colon M\to \mathbb{R}$ be a $C^2$, geodesically convex function which has $L$-Lipschitz-...
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Is the square root of the Kullback-Leibler divergence a convex map?
$\newcommand{\KL}{\operatorname{KL}}$Let $X$ be a Polish metric space and $P(X)$ the space of probability measures on $X$. Given $\mu, \nu\in P(X)$, recall that
$$\KL(\mu\parallel\nu) = \begin{cases}\...
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0
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263
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Starlike sets in $\mathbb{C}^n$
Let $S$ be a bounded domain in $\mathbb{C}^n$. $S$ is called starlike about the point $x_0\in S$ if for every point of $S$, the segment of the straight line from the point to $x_0$ lies in $S$. If $S$ ...
2
votes
2
answers
132
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Convexity of the exponential of the negative Renyi entropy
I would like to try my luck here for the following question after failing to elicit an answer to it on math.stackexchange.com.
For $r\ge -1$, the exponential of the negative Renyi entropy is defined ...
4
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144
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How many convex or concave subsets are contained in an arbitrary set of $n$ real numbers?
This question is closely related to this post. A set $A=\{a_1<a_2<\dots<a_n\} \subset \mathbb R$ is said to be convex if the consecutive differences are non-decreasing, i.e. if $a_{j+1} - a_j ...
1
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1
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97
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minimum eigenvalue interpolation
Suppose we have two symmetric positive definite matrices $A,B$ (not simultaneously diagonalizable). How can I find a matrix function $f(t), t\in [0,1]$, such that $f(0)=A, f(1)=B$, and the minimum ...
4
votes
1
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114
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Is $0$ a member of the following special kind of a convex compact set?
Let $(V, \lVert \cdot \rVert)$ be a normed space. Let us consider the set $C = [-1,1]^{\dim V}$. The boundary of this set consists of closed subsets $B_i$ (indexed by some set $I$) of affine ...
1
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0
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58
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Minimum bounding rectangle of symmetric convex bodies in the plane : is the ball the worst case
The minimal rectangle containing the euclidean ball in the plane is the standard cube $B_\infty = [-1;1]^2$. I would like to know if the euclidean ball is the worst symmetric convex body to be ...
2
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0
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79
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A convex version of the small uncountable cardinal $\mathfrak b$
Let us recall that $\mathfrak b$ is the smallest cardinality of a subset of $\omega^\omega$, which cannot be covered by countably many compact subsets of $\omega^\omega$.
The definition of $\mathfrak ...
2
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0
answers
59
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Does absolute retract imply convex structure?
In the theory of selection, it is known that any compact absolute retract (AR) carries a convexity structure defined by E. Michael. It is also known that a convex structure
developed by Van de Vel ...
5
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0
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208
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Reference request for convex geometry?
I am looking for a reference for an elementary convex geometry.
In Appendix A (page 1810) of this paper by Green and Tao, they cover some basic results from elementary convex geometry. The results ...
17
votes
1
answer
328
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Functions of $\mathbb{R}^d$ preserving convexity of sets
Consider a function $f : \mathbb{R}^d \to \mathbb{R}^d$, with $d\geq 2$, such that:
$f$ is injective,
For any convex set $A$ of $\mathbb{R}^d$, $f(A)$ is also convex.
What can we say about $f$ ? In ...
5
votes
2
answers
242
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Which convex subsets of a normed space are intersections of balls?
Let $(V, \lVert \cdot \rVert)$ be a normed space. For any $A \subseteq V$, let $O(A)$ be the intersection of all closed balls containing $A$, or more precisely, let $O \colon 2^V \to 2^V$ be defined ...
3
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0
answers
84
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Projection onto level set of convex functional
Fix a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and let $F:L^2_{\mathbb{P}}(\mathcal{F})\rightarrow (-\infty,\infty]$ be bounded-blow, convex, lower semi-continuous, and not identically ...