# Questions tagged [convexity]

For questions involving the concept of convexity

559
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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(\...

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### Some simple conditions on a function $f$ so that $x\mapsto xf(x)$ is convex?

Studying the Braess Paradox for a project at school (with the assiociated Wardrop's equilibria and Nash's game theory result) I came upon one simple question I can not figure out….
If $f$ is a ...

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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 ...

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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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### Using the duality theorem for an optimization problem with two variables

I wish to apply the duality theorem to the optimization problem:
$$\text{minimize}~~s$$
$$\text{subject to}~~g_j(x)\leq{s},~~\text{for all}~~j=1,...,r,~~x\in{X},~~s\in{\mathbb{R}},$$
where $X\subseteq{...

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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 ...

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### Is this function concave? If it is, can we show it in a theoretical way?

Suppose we have a function:
\begin{equation}
\begin{aligned}
f(x_1,x_2,\cdots,x_n)&=\sum\limits_{i=0}^n\frac{(x_i e^{-x_i}-x_{i+1}e^{-x_{i+1}})^2}{e^{-x_i}-e^{-x_{i+1}}}\\
&=\frac{(x_0 e^{-x_0}...

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### Lovasz extension and underlying matroid

Let $F \colon 2^V \rightarrow \mathbf{R}$ be a set function for some ground set $V = \{ 1, \dots, n\}$. The domain of $F$ is $\{ 0,1\}^n$ , under the usual identification of a set $S$ with its ...

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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 ...

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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
$$ \frac{\partial^2}{\partial x_1^2}u <0 $$
...

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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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### 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 ...

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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 ...

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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 ...

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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 $f(x) - f(x^*) \leq (const) \cdot L r$ 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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### 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 $S$ ...

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### Projection onto a cone followed by a Schur-convex function

Let $Proj_C(x)$ denote the projection of a point $x$ onto a cone $C$. Let $f$ be a Schur-convex function. I'm considering $f(Proj_C(x))$ as a function of $x$.
Are there any conditions on the cone $C$ ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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### Convexification of difference of convex functions

I am looking for a reference/a hint to the following problem:
We are given $f_1(x),f_2(x)$ convex functions (say, on $\mathbb R^d$) such that $f_1(x) \to\infty$ for $\|x\|\to\infty$. Also there is an $...

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### Strange result about convexity

$f \in C^2([0,1])$ with $f''$ convex and $f(0) = f'(0) = f''(0) = 0$.
Is it true that : $f''(1)+6f(1)\geq 4f'(1)$ ?
Source: AoPS

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### Maximal geodesically convex function interpolating three points on the hyperbolic plane

Crossposted on MSE: https://math.stackexchange.com/questions/4282998/maximal-geodesically-convex-function-interpolating-three-points-on-the-hyperboli
Let $M$ be a two-dimensional Hadamard manifold. ...

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### Does the ball maximize the "kissing probability" of symmetric convex bodies? [duplicate]

Given a symmetric convex body $K \subset \mathbb{R}^n$ (i.e., a bounded symmetric convex set with non-empty interior), I am interested in the following quantity
$$p_K := \Pr_{x_1, x_2 \sim K}[x_1 \in ...

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### How to prove the inequality $\ln\frac{1+e^{-y}}{1+e^{-x}}+\frac{1}{1+e^x}(y-x)\geq 0$?

I am trying to prove that $\ell(\beta) = \sum_{i=1}^n \left (-y_i \beta^{\top}x_i + \ln \left (1 + e^{\beta^{\top}x_i }\right )\right )$ is a convex function. I follow the following steps:
Let $\...

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### Convexity of $(X, y) \mapsto y^T X^{-1} y$ [closed]

Let $y \in \mathbb{R}^n$, $X \in \mathcal{S}^n_{++}(\mathbb{R})$. Why would function $ f : (X, y) \mapsto y^T X^{-1} y$ be convex?
I tried with $(X, x) + t.(Y, y)$ with no result. Also, I thought ...

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### Characterization of inverse limits of finite-dimensional convex cones

Consider a countable inverse system $C_1\substack{f_1 \\ \leftarrow} C_2 \substack{f_2 \\ \leftarrow} C_3 \substack{f_3 \\ \leftarrow} \ldots$ where the $C_i$ are finite-dimensional convex cones of ...

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### When is a continuous subadditive function (0,1]-superhomogeneous

Continuous version of this Superhomogeneity of subadditive functions
Let $f$ be a continuous function of $\geq 2$ real variables defined on a convex cone $\mathcal{C}$ in the upper half plane, with $f(...

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### Superhomogeneity of subadditive functions

Let $f$ be a function of $\geq 2$ real variables defined on a convex cone $\mathcal{C}$ in the upper half plane, with $f(0) = 0$. Suppose $f$ is subadditive, i.e. $f(x_1+y_1, \dots,
x_n+y_n) \leq f(...

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### Log-concavity of lattice-functions and convolution

I was looking at the definition of log-concavity:
A function $F:\mathbb{R}^n\rightarrow\mathbb{R}$ is said log-concave
iff $F(x)\geq 0\forall x\in\mathbb{R}^n$ and
$$F^\lambda(x)F^{1-\lambda}\leq F(\...

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### Worst convex compact set for translational packings of $\mathbb R^d$

A (translational) packing of a convex compact subset (with non-empty interior) $\mathcal C$ of $\mathbb R^d$ is a union
of translated non-overlapping (but perhaps touching) copies
of $\mathcal C$.
The ...

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### Log-concavity of the modified Bessel function of a second kind

I was searching for some results for the log-concavity of the modified Bessel function of a second type, but I failed. Has there been any known work on this? I am not even sure if it is the modified ...

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### Inscribed square and convexity

Let $b(X)$ be the boundary of any $X$ subset of the plane.
Does there exist $A,B$ convex compact sets of the plane, such that $C:=A\setminus B$ is simply connected and not empty, and such that ...

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### Length of curves on convex hypersurfaces

Let $\gamma\colon[a,b] \to \mathbb{R}^n$ be a smooth curve.
Let $f_i\to f$ be a sequence of convex functions on $\mathbb{R}^n$ converging uniformly on compact subsets to $f$.
Let $\hat\gamma(t):=(\...

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### Busemann-Feller lemma in hyperbolic space

The classical Busemann-Feller lemma in Euclidean space says the following.
Let $K\subset \mathbb{R}^n$ be a closed convex set.
Then
for any point $x\in \mathbb{R}^n$ there exists unique nearest point ...

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### How to prove the convexity of a simple function involving a ratio of two polygamma functions?

Let
\begin{equation*}
\Gamma(z)=\int_0^{\infty}t^{z-1}\textrm{e}^{-t}\textrm{d}t, \quad \Re(z)>0
\end{equation*}
and
$$
\psi(z)=[\ln\Gamma(z)]'=\frac{\Gamma'(z)}{\Gamma(z)}.
$$
In the literature, ...

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### Factorizing the doubly stochastic matrix where all entries are equal such that the factors are all convex combinations of few permutation matrices

Let $N_{n}=(1/n)_{i=1,j=1}^{n}$ be the $n\times n$-matrix where all the entries are equal.
Suppose $n>0$. Let $\delta_{n}$ be the least natural number such that $N_{n}$ can be factored as $N_{n}=A_{...

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### Can the subdifferential become unbounded at interior points?

Consider $f: \mathbb{R}^n \to \overline{\mathbb{R}}$ a lower-semicontinuous, proper, closed and convex. My question is, can the subdifferential of $\partial f$ be unbounded in the interior of $\text{...

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### Biconjugate of a quasiconvex lower semi-continuous function

Let $f:\mathbb{R}^d \to [0,\infty]$ be a quasiconvex lower semi-continuous function whose effective domain $C:=\{x \in \mathbb{R}^d:f(x) < \infty\}$ is nonempty and bounded (and convex since $f$ is ...

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### Convexity of a set of probability densities

Consider the space of probability densities $(P(\mathbb{R}^d), W_2)$ (probability measures on $\mathbb{R}^d$ with 2-Wasserstein distance).
How can we determine if a subset $Q$ is convex?
I know that a ...