Questions tagged [convexity]

For questions involving the concept of convexity

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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, $c: X \times Y \rightarrow \mathbb{R}$, ...
Student's user avatar
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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)\...
user_XL's user avatar
2 votes
2 answers
183 views

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 ...
Joshua Isralowitz's user avatar
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1 answer
121 views

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 ...
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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 $E_u$ ...
User1999's user avatar
6 votes
2 answers
300 views

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 ...
Alexander Pruss's user avatar
20 votes
3 answers
1k views

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 ...
Piotr Hajlasz's user avatar
7 votes
0 answers
109 views

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\...
alesia's user avatar
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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 \...
Reza's user avatar
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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\...
G. Panel's user avatar
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1 answer
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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 ...
user19904's user avatar
1 vote
1 answer
148 views

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 ...
Ali's user avatar
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2 votes
1 answer
322 views

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 ...
Ayman Moussa's user avatar
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2 answers
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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 ...
Aaron Goldsmith's user avatar
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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 ...
Shih-Chi Liao's user avatar
7 votes
2 answers
438 views

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 ...
AlwaysLost123's user avatar
2 votes
1 answer
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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:...
Nik Bren's user avatar
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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{...
G. Panel's user avatar
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4 votes
2 answers
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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 $\partial C$. ...
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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 ...
guest61's user avatar
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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?
Sounak's user avatar
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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= \...
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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 ...
Gericault's user avatar
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1 answer
158 views

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'...
Steve's user avatar
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6 votes
1 answer
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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 ...
Denis Serre's user avatar
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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(\...
Cryme's user avatar
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2 votes
1 answer
275 views

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 ...
Kacper Kurowski's user avatar
7 votes
2 answers
530 views

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 ...
MathArt's user avatar
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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 ...
Tryer's user avatar
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3 votes
0 answers
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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 ...
ARG's user avatar
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4 votes
2 answers
260 views

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 $$ \frac{\partial^2}{\partial x_1^2}u <0 $$ ...
Ali's user avatar
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0 votes
0 answers
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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 ...
user332905's user avatar
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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 ...
user332905's user avatar
3 votes
1 answer
616 views

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 ...
iolo's user avatar
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4 votes
2 answers
207 views

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 ...
Luis Yanka Annalisc's user avatar
3 votes
1 answer
231 views

$\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^...
Dattier's user avatar
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5 votes
1 answer
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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-...
ccriscitiello's user avatar
8 votes
1 answer
658 views

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}\...
ECL's user avatar
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1 vote
0 answers
263 views

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 $S$ ...
user332912's user avatar
2 votes
2 answers
132 views

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 ...
Hans's user avatar
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4 votes
0 answers
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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 ...
Oliver Roche-Newton's user avatar
1 vote
1 answer
97 views

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 ...
Mikhail's user avatar
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4 votes
1 answer
114 views

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 ...
Kacper Kurowski's user avatar
1 vote
0 answers
58 views

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 ...
Gericault's user avatar
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2 votes
0 answers
79 views

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 ...
Taras Banakh's user avatar
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2 votes
0 answers
59 views

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 ...
Shijie Gu's user avatar
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5 votes
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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 ...
Johnny T.'s user avatar
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17 votes
1 answer
328 views

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 ...
Pohoua's user avatar
  • 303
5 votes
2 answers
242 views

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 ...
Kacper Kurowski's user avatar
3 votes
0 answers
84 views

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 ...
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