Questions tagged [convexity]
For questions involving the concept of convexity
665 questions
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Are geodesically convex sets closed under intersection and diffeomorphic to $\mathbb R^n$?
Bott and Tu's book on algebraic topology proves the existence of good covers on smooth manifolds by a notion on differential topology:
Now we quote the theorem in differential geometry that every ...
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154
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Strict convexity, continuous modulus, and Kadets-Klee property
Let $(X,\|\cdot\|)$ be a Banach space. Assume that $X$ is strictly convex and that its modulus of convexity
$$
\delta_X(\varepsilon)=\inf\left\{1-\left\|\frac{x+y}{2}\right\| \colon \|x\|=\|y\|=1,\ \|...
- 341
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Literature for checking a function is c-concave
I would like to know if there is some update about checking a given function is $c$-concave, especially when $c$ is not (strongly) convex. Here we say $f$ is $c$-concave if $f$ is the $c$-transform of ...
- 231
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Convexity principle in several complex variables
I would like a reference for an analogue of the Phragmen-Lindelof in several complex variables. Specifically, if f is analytic in a region $A \subset \mathbb C^n$ and, by Bochner's theorem it extends ...
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169
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Extreme points of a certain compact convex set
Say $\mathfrak{A}$ is a seperable $C^*$ algebra, the space $K$ of states on $\mathfrak{A}$ is compact and convex. Let $\Gamma$ be a countable discrete group acting on $\mathfrak{A}$ via $*$-hom. This ...
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Convex concentration of a tensor-squared spherical vector
Let $X$ be a random vector in $\mathbb{R}^n$. We say that $X$ has the convex concentration property with constant $K > 0$ if for every $1$-Lipschitz convex function $\varphi : \mathbb{R}^n \to \...
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Vanishing of Monge–Ampère operator on a convex function
Let $\Omega_1,\Omega_2\subset\mathbb{R}^n$ be strictly convex bounded domains, possibly with smooth boundary, and such that $\bar\Omega_1\subset\Omega_2$.
Let $u\colon \bar\Omega_1\to \mathbb{R}$ be a ...
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Log concavity of a Gaussian function
Fix $t > 0$ and consider the map
$$
f(x) = \log \mathbb{P}\{|\sqrt{x} + Z| \leq t\},
$$
where $Z$ is a standard Normal random variable on the real line.
Is it true that $f$ is concave on the ...
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A general result on the convex function
During my research, I came a cross this question :
Let $f \in C([0,1])$ convex.
Is it true that $$\int_0^1 \max(f(t),f(1-t)) \geq\\ 2/3\max(f(1/3),f(2/3))+1/3 \max(f(1/6),f(5/6))?$$
- 6,010
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All possible images of a vector after averaging
For a given $x \in \mathbb{R}^n$, is there a name given for the set of all possible $y\in \mathbb{R}^n$ that can be obtained by iteratively selecting a subset of entries $S\in \{1,\dots,n\}$ and ...
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Proof of skew inequality for convex functions of Gaussian measure?
Let $X$ be a standard $n$-dimensional Gaussian random variable, and $f \colon \mathbb{R}^n \to \mathbb{R}$.
It is known (see Remark 2.4.2 in this paper) that if $f$ is convex, then
$$
\mathbb{E} f(X) \...
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Truth of a conjecture about the largest incircle of convex polygons
The problem of determining the largest incircle of a convex polygon can be solved by constructing the Voronoi diagram of the polygon's edges and selecting the vertex of the diagram's graph with ...
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An inequality implication for shifted sums and products
I’m working on a problem that boils down to the following clean special case. Fix positive integers $b_1, b_2, c_1, c_2 $ and integer $k > 2$. I am attempting to get a concise, self-contained proof ...
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How to design encoders with the minimum number of rows?
Assume that matrices $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}=\mathbf{A}\mathbf{B}$ are given. I aim to find matrices $\mathbf{E}_1$, $\mathbf{E}_2$, and $\mathbf{E}_3$ such that
\begin{align}
\...
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Subdifferential $\partial \varphi(u)$ admits a measurable selection, where $\varphi:\mathbb{R}^3\to \mathbb{R}$ is convex and $u$ is measurable
I am looking for a reference for the following claim:
Let $\Omega\subset \mathbb{R}^3$ be a bounded domain, $u:\Omega\to \mathbb{R}^3$ be measurable and let $\varphi:\mathbb{R}^3 \to \mathbb{R}$ be ...
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Characterization of convex positively homogeneous functions according to behavior on unit sphere
For now just speaking in $\mathbb{R}^2$ we call a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ positively homogeneous if for every $\lambda \ge 0$ we have $f(\lambda x )=\lambda f(x)$. For such a ...
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What is the closed convex hull of convex ridge functions?
Crossposted at Mathematics SE
Let us define the following class of functions:
$$ \mathcal{F} := \left\{ f : \mathbb{R}^d \to \mathbb{R} \,\middle|\, f(x) = \sum_{i=1}^n \varphi_i(\xi_i^\top x),\ \...
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A particular continuous selection problem
Let $B$ and $S:=\partial B$ denote, respectively, the unit ball and the unit sphere wrt to some norm $\|\cdot\|$ on $\Bbb R^2$. For a fixed real $r\in(0,1]$ and all $u\in S$, let
$$D_u:=(B+ru)\cap(B-...
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A continuous selection problem
Let $(B_t)_{t\ge0}$ be a continuous (say wrt the Hausdorff metric) family of nonempty centrally symmetric convex compact subsets of $\Bbb R^2$. For a norm $\|\cdot\|$ on $\Bbb R^2$ and each real $t\...
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Convex conjugate of sum of functions of overlapping pairs of variables
Let $ f: (0, \infty)^{2} \mapsto (0, \infty) $ smooth, non-negative, convex. Define $F(x_1, x_2, x_3):(0, \infty)^{3} \rightarrow [0, \infty) = f(x_1, x_2) + f(x_1, x_3) + f(x_2, x_3)$.
How to express ...
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The spaces $L^p(0,1), 0<p<1$ have the needle property
It seems that J. W. Roberts in his article "Pathological compact convex sets in the spaces $L_p, 0<p<1$" has proved the following property.
For every $\varepsilon>0$ there exists a ...
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How can we show an upper bound on the Bregman divergence from strong convexity?
So I know that strong convexity for a function gives a lower bound on the Bregman divergence based on the squared norm of the difference between two points $x$ and $y$, but apparently it also implies ...
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Approximation of log-concave distribution by distribution of weighted sum of exponential r.v.s
In a conjecture from this question, the exponential distribution, probably, is a corner case among log-concave absolutely continuous probability distributions.
Let $n \in \mathbb{N}$, let $a_0, a_1, \...
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A special convex cone of convex functions
This question is related to On the convex cone of convex functions
Let $\Omega \subset \mathbb R^d$ be a closed set. Define $\mathcal F$ to be the set of (continuous) convex functions on $\Omega$, and ...
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Question about the proof of John's Theorem from Convex Geometry
I am trying to understand the proof of John's theorem from The John Ellipsoid Theorem by
Ralph Howard, which asserts that if $K\subset\mathbb{R}^n$ is a convex body, then there exists an ellipsoid $\...
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Gradient norm of convex function restricted to points bounded away from argmin is minimized at the boundary of the restriction
Say $f: \mathbb{R}^{d} \to \mathbb{R}$ is convex and differentiable everywhere. Suppose also that $f$ is minimizable, and denote $argmin f := S$ (we may assume $S$ is bounded). Pick any $\delta >0$....
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Some kind of scaled convexity
I have the following discrepancy that satisfies this type of inequality. I want to know if this can be related to some kind of weak convexity. If so, what is the name of this property? Additionally, ...
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Existence of regular bounded self-maps on a Banach space
Which Banach spaces $X$ admit a bounded function $\phi:X\to X$
of given regularity such that
(a) $\phi(x)=x+o(x)$ for $x\to0$,
or even
(b) $\phi(x)=x$ in a neighbourhood of $0$
$$\bf ?$$
There always ...
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A local sufficient condition for the superadditivity that is strictly more general than the convexity
$\newcommand\R{\Bbb R}$Let $(S,+)$ be a semigroup. A function $f\colon S\to\R$ is superadditive if $f(x+y)\ge f(x)+f(y)$ for all $x$ and $y$ in $S$.
For instance, any convex function $f\colon[0,\infty)...
- 144k
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Convexity or concavity of this weighted distance
I am working with a weighted distance between probability distributions and want to understand it better. Assume that $X$ and $Y$ are two random variables with a joint probability mass function $p_{X,...
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Which positive convex cones in $\mathbb{R}^{n}$ are closed under componentwise meets?
The vector space $\mathbb{R}^{n}$ has a natural lattice structure: for $\mathbf{a} = (a_1, \dots, a_n)$ and $\mathbf{b} = (b_1, \dots, b_n)$
$\mathbf{a} \wedge \mathbf{b} = (\min(a_1,b_1), \dots, \min(...
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Maximal property of c-cyclically monotone sets
Let $X$ and $Y$ be finite sets and $c:X\times Y\rightarrow \mathbb{R}$ be a real cost function. A subset $\Gamma\subseteq X\times Y$ is $c$-cyclically monotone if for all $n$ and $\{(x_i,y_i)\}_{i=1}^...
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A question on Ibragimov's theorem on strong unimodality
I am not a mathematics student and unfortunately have some confusion about a (well-known) theorem about strong unimodality of distributions. First of all let me clarify some terminologies and then ask ...
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On the convex cone of convex functions
$\newcommand\R{\Bbb R}$Let $F$ be the set of all functions of the form $\max(a,b,c)$, where $a,b,c$ are affine functions from $\R^2$ to $\R$ and the maximum is taken pointwise. Let $G$ be the set of ...
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Why cannot we adapt Barvinok type counting techniques to general convex integer programs?
Decision problems in Integer Linear Programming have Lenstra type algorithms (https://www.math.leidenuniv.nl/~lenstrahw/PUBLICATIONS/1983i/art.pdf) have been generalized to convex integer program ...
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What are the injective embeddings of R^d into the cone of (semi-) positive definite matrices of dimension d?
How can we characterize the set of all injective functions from $\mathbb{R}^d$ to the set of all symmetric positive definite matrices of dimension d?
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An s-convex function lying between two convex functions
Let $f: \mathbb R_{+} \to \mathbb R_{+}$ be an $s$-function in the second sense, i.e.,
$$ f(\lambda x +(1-\lambda)y) \leq \lambda^s f(x) +(1-\lambda)^s f(y)$$ for every $\lambda \in (0,1)$. Assume ...
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vote
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Generalization of subadditivity analogous to quasiconvexity, and variants
I am curious if there are natural generalizations of subadditivity which have been studied in the past or have been stated in the literature? I (and people that I have talked to) have not had much ...
- 21
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546
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Maximum and concavity of function
Let
\begin{align}
G(x_1,x_2,x_3)=\int_0^{2\pi}\int_0^{2\pi}\int_0^{2\pi} \frac{d\theta_1 d\theta_2 d\theta_3}{(2\pi)^3} p(\vec{x},\vec{\theta}) \log\left( p(\vec{x}, \vec{\theta})\right)
\end{...
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Reference request: books on convex analysis / geometry
I am interested in convex geometry and analysis, especially in its connections with high dimensional probability theory.
I was reading the book by Pisier, The volume of convex bodies and Banach space ...
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Is the norm of first or second level of of signature a convex function?
I understand this is not a research level question but I really want to know, would anyone please help.
This question is related to the signatures that arises in rough path theory.
Is there any ...
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Limits along lines for the gradient of a convex function
It is easy to see that if a function $f: \mathbb{R} \to \mathbb{R}$ is strictly convex, $C^1$ and $f'$ has bounded image, then as $t\to \infty$ the limit
$$
\lim_{t\to\infty} f'(t) = \lim_{t\to\infty} ...
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Is there a path-connected, "anti-convex" subset of $\mathbb R^2$ containing $(\mathbb R\smallsetminus \mathbb Q)^2$?
This question was firstly asked in mathematics stack exchange. Getting no answer, I copied it to here.
For a vector space $V$ over $\mathbb R$, I say a subset $S$ of $V$ is "anti-convex" if $...
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97
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Strengthening the Kovner-Besicovich theorem: Does every unit-area convex set in the plane contain a centrally symmetric hexagon of area $2/3$?
The Kovner-Besicovich theorem states that every convex set $S$ in the plane contains a centrally symmetric subset $C$ of at least $2/3$ the area of $S$, and that this bound is sharp for triangular $S$....
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131
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Do separable cubic constraint and separable quartic constraint SOCP presentable?
I am an engineer who is doing some network modeling and optimization. During my work, I was running into a case that is quite strange. The problem that I am trying to solve seems to be convex and it ...
7
votes
2
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428
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Integral means vs infinite convex combinations
Let $(X,\mathcal A, \mu)$ be a probability space, $\mathbb E$ a Banach space, and $f:X\to\mathbb E$ a Bochner integrable function.
Does there exist a sequence $(x_k)_{k\ge 1} $ in $X$, and a ...
- 63.5k
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0
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78
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Construct compact submanifold containing non-compact Nash embedded submanifold
$$
\newcommand{\R}{\mathbb{R}}
\newcommand{\geu}{g_{\text{Eu}}}
\newcommand{\X}{\mathcal{X}}
\newcommand{\iX}{\mathring{\X}}$$
Let $\X$ be a closed bounded convex set in some Euclidean space. Its ...
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votes
1
answer
234
views
Proving convexity of the expected logarithm of binomial distribution
I would like to prove that the following function, for an arbitrary integer $n$:
\begin{equation}
\begin{split}
f(x) & =x\cdot E \ \log(1+\text{Binomial(n,x)}) \\
& = x \cdot \sum_{k=0}^{n} \...
- 23
1
vote
0
answers
46
views
Finite right-triple convex sets in planes
Let $\mathcal{S}$ be a set of points in $\mathbb{R}^2$. We say that $\mathcal{S}$ is right-angle convex, if for any two distinct points $P,Q\in \mathcal{S}$ there always exists another point $R\in \...
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1
answer
242
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Does there exists an example of a Banach space that is compactly LUR; but not LUR
We know that a Banach space $X$ is locally uniformly rotund(LUR) if for $x, x_n \in S_X$ with $\Vert x+x_n \Vert \to 2$, we have $x_n \to x$. In the same case, if $(x_n)$ has a convergent subsequence, ...
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