Questions tagged [convex-analysis]
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507
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A potential new norm for matrices and Horn's inequalities
I am investigating a function defined in terms of the singular values of matrices. Initially, I simplified the problem by focusing on the eigenvalues of $2 \times 2$ Hermitian, positive-definite ...
0
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0
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82
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Generalization of Kakutani-Ky Fan Theorem without convexity assumptions
Crossposted at Mathematics SE
I'm wondering if there exists some extension of the Kakutani-Ky Fan Theorem
Theorem. Let $K$ be a nonempty, compact and convex subset of a locally convex space $X$. ...
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3
votes
1
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201
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Interesting question about the Thomson problem for arbitrary number of electrons
This question is crossposted from here I believe this is a pretty hard question and so I decided to repost the question in the Math Overflow forum. If there is something wrong with doing this, I am ...
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1
vote
1
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46
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From relative convexity to modulus of continuity estimates for the dual gradient mapping
Let $F: \mathbf{R}^d \to \mathbf{R}$ be a convex function, let $m > 0$, and define $Q_m: \mathbf{R}^d \to \mathbf{R}$ to be the mapping $x \mapsto \frac{m}{2} \| x \|_2^2$. One says that $F$ is $m$-...
- 706
2
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1
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112
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Smooth approximation of nonnegative, nondecreasing, concave functions
Let $f\colon [0, \infty)\to\mathbb{R}$ be nonnegative, nondecreasing, and concave. Prove the following claim or give a counter example: There is a sequence of functions $f_n\colon [0, \infty)\to\...
- 63
18
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3
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919
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Convex functions in convex sets
Suppose $\Omega \subset \mathbb{R}^n$ is some bounded, convex set. For which domains $\Omega$ is it true that for every convex function $f:\Omega \rightarrow \mathbb{R}$ the average of the function in ...
3
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2
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189
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Subdifferential of a convex function admits a continuous selection
Let $F$ be a continuous convex function on $\mathbb{R}^n$.
If the subdifferential $\partial F(x)$ of $F(x)$ admits a continuous selection, for every $x \in \mathbb{R}^n$, does it mean that $F$ is ...
- 33
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0
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110
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Characterization of a "complex" hull?
This is a complex continuation of my previous question. There Iosif Pinelis showed that the so obtained closure from taking the intersection of the preimages of the linear functionals indeed coincides ...
- 6,718
1
vote
1
answer
66
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Viscosity characterization of convex functions
Let $\Omega\subseteq\mathbb{R}^n$ open and convex. It is elementary that if $u\in C^2(\Omega)$ then
$$u \text{ is convex}\iff D^2u\geq0 \ \text{ in } \Omega$$
I was looking for a similar ...
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6
votes
2
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535
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Conditions for including cones
Consider $N$ $n$-dimensional vectors, where the angle between any two vectors is acute and their starting point is at the origin. Can we rotate these vectors together so that the coordinate components ...
- 61
2
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1
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321
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Tangent cone of a closed convex cone
Let $K \subset \mathbb{R}^n$ be a closed convex set. Given a point $u \in K$, the tangent cone of $K$ at $u$ is defined as (or characterized by)
$$
T_K(u) := \mathrm{cl}(\left\{ t (v - u) \mid v \in K,...
- 153
4
votes
2
answers
224
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An upper bound of gradient norm for convex functions near minimizer
Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a convex function. Denote $X^*$ as the set of minimizers of $f$ and assume $X^*$ is unbounded. Is it possible that $\|g_x\|$ is unbounded when $d(x,X^*)$ ...
- 311
1
vote
1
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80
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Legendre transformation of vector valued function
Good afternoon. Is there any generalisation of Legendre--Fenchel transformation to the vector-valued functions $f: \mathbb{R}^n \to \mathbb{R}^n$?
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1
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Large ideally convex sets
Let $E$ be a Banach space. A set $C \subseteq E$ is called ideally convex if for every bounded sequence $(x_n)$ in $C$ and for every sequence $(\lambda_n)$ in $[0,1]$ that sums up to $1$ the vector $\...
- 11.6k
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0
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On convexity of special fractals in the plane
Let $X \subset \mathbb{R}^2$ be a subset of the plane with Hausdorff dimension $1<dim_H(X)<2$.
For a subset $Y \subset \mathbb{R}^n$ we define $Y$ to be convex if for every $y_1,y_2 \in Y$ the ...
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1
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Asking for an English version of Aleksandrov's famous 1939 paper in Convex Geometry
I have difficulty even in finding a Russian version of the next paper:
"Aleksandrov, A. D., Almost everywhere existence of the second differential of a convex
function and some properties of ...
- 290
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0
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101
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The study of directional derivatives for functions that are minimums of convex functions
Has research been conducted on the topic of directional derivatives of functions that are minimums of convex functions? It would be greatly appreciated if you could provide the appropriate reference.
...
2
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Whether $d_x(t) := \|P_t(x)-x\|_H$ is increasing in $t$ where $P_t:H \to H$ is the proximal operator of a convex function
Let $H$ be a Hilbert space (e.g Euclidean $\mathbb R^n$), and fix a proper convex function $f:H \to (-\infty,+\infty]$. Given any $t \ge 0$, let $P_t:H \to H$ be the proximal operator of $f$ at level $...
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0
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Support function of the intersection of a hyper-ellipsoid and a Euclidean ball
Le $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_d$ be positive numbers. For any $x \in \mathbb R^d$ and $r \ge 0$, define $\gamma(x,r) := \sup_{z \in E(r)}x^\top z$, where
$$
E(r) := E \cap B_2^d(r)...
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0
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158
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Bound for the laplacian of a strictly convex function from above by the gradient of it
Let $V \in C^2(\mathbb{R}^d; \mathbb{R})$ a (strictly) convex function with $ \int_{\mathbb{R}^d} \mathrm{e}^{-V(x)} \, dx = 1.$
I am trying to show that
$$ \int_{\mathbb{R}^d} |\nabla_x V(x) |^2\...
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0
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99
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Convex minorants to convex functions, given partial Taylor expansion and smoothness estimate
Let $V$ denote a strictly convex function (in arbitrary dimension) whose Hessian is $L$-Lipschitz. Given only this knowledge, and the values of $\left\{ V \left( x \right), \nabla V \left( x \right), \...
- 706
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1
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Strict convexity of the cost function is enough to ensure the existence and uniqueness of the optimal transport map
Let $X=Y = \mathbb R^d$ and $c:X \times Y \to [0, \infty)$ be Borel measurable. Let $\mu, \nu$ be Borel probability measures on $X,Y$ respectively. Let $\mathcal T$ be the set of all Borel measurable ...
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1
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Is a Lipschitz continuous gradient equivalent to this condition?
I know if a function $f: \mathbb{R}^n \to \mathbb{R}$ is $L$-smooth, i.e. its gradient $\nabla f$ is $L$-Lipschitz continuous, then it satisfies the following inequality for any $x, x_0 \in \mathbb{R}^...
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1
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Is the Legendre transform as an operator Lipschitz?
Let $C_{lsc}(\mathbb{R}^n)$ be the space of lower semicontinuous convex functions $\mathbb{R}^n \to \mathbb{R}$. The Legendre-Fenchel (LF) transform of $f \in C_{lsc}(\mathbb{R}^n)$ is:
$$ f^*(y) := \...
- 161
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0
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130
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A Lipschitzian's condition for the measure of nonconvexity
I'm actually working on the measure of nonconvexity and its application. Especially, the Eisenfeld–Lakshmikantham MNC defined - in a Banach space - by:
$$\alpha(A)=\sup_{b\in\overline{\operatorname{...
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Are there any characterizations of $C^2$ convex functions?
There are several characterizations of convex functions with the Lipschitz continuous gradient. If we already know that the function is of class $C^1$, then we have the following equivalent conditions:...
- 27.3k
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1
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124
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Strict inclusion for recession cone of closure of a convex set
Let $C$ be a nonempty closed convex subset of $\mathbb{R}^n$. The recession cone of $C$ is given by
$$R_C=\left\lbrace d\in\mathbb{R}^n:x+td\in C, \forall t>0, \forall x\in C\right\rbrace.$$
It is ...
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1
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0
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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$ ...
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On the infimal convolution of two norms on $\mathbb R^n$
$\newcommand{\R}{\mathbb R}$For natural $n$, $a\in\R^n$, and real $t>0$, let
\begin{equation*}
K:=K_{n,t}(a):=\inf_{x\in\R^n}(\|a-x\|_2+t\|x\|_1),
\end{equation*}
\begin{equation*}
M:=M_{n,...
- 118k
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Is the mapping $F(a):= \arg\min_{x \in \mathbb R^n} \|x-a\|_2 + \|x\|_1$ non-expansive?
Fix $a \in \mathbb R^n$ and let $\|\cdot\|$ be any norm on $\mathbb R$ (e.g $\ell_1$ norm). For any $a \in \mathbb R^n$, it is clear that the function $f_a(x) := \|x-a\|_2 + \|x\|$ is strictly convex ...
- 6,726
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1
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Orthogonal projection of a point centrally-symmetric closed convex subset of $\mathbb R^n$ never expands the coordinates of the point
Let $C$ be a closed convex subset of $\mathbb R^n$ which is symmetric about the standard coordinate axes. For example, think of $C$ as the unit-ball for an $\ell_p$-norm, for some $p \in [1,\infty]$. ...
- 6,726
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0
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Minimax statistical estimation of proximal transform $\mbox{prox}_g(\theta_0)$, from linear model data $y_i := x_i^\top \theta_0 + \epsilon_i$
tl;dr: My question pertains the subject of minimax estimation theory (mathematical statistics), in the context of linear regression.
Given a vector $\theta_0 \in \mathbb R^d$, consider the linear ...
- 6,726
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0
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Any reference on Jensen inequality for measurable convex functions on a Hausdorff space?
I asked this question on math.stackexchange and I was suggested that asking it may be more appropriate. This is part of my research which tries to extend some of Choquet's theory to some non-compact ...
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2
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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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1
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Analytic value of $\alpha := \sup_{(x,y) \in C} ax+by$, where $C := \{(x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$
Let $a,b \in \mathbb R$, $R \ge 0$, and $c > 0$. Define $C := \{(x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$, and set
$$
\alpha := \sup_{(x,y) \in C} ax + b y.
$$
Question. ...
- 6,726
0
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1
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Analytic formula for minimizer of $f(x) := \sqrt{(x-a)^\top S(x-a)}+ r \|x\|_2$
Let $S$ be a positive-definite $n \times n$ matrix and define $\|z\|_S := \sqrt{x^\top S x}$ for any $x \in \mathbb R^n$. Let $a$ be a fixed vector in $\mathbb R^n$ and $r \ge 0$, and consider the ...
- 6,726
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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 ...
- 27.3k
3
votes
1
answer
191
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Probabilistic Taylor theorem for concave functions
This paper proves a probabilistic version of Taylor's theorem
\begin{equation*}
\mathbb{E}g(X) = \sum_{k=0}^{n-1} \frac{g^{(k)}(0)}{k!} \mathbb{E}X^k + \frac{\mathbb{E}X^n}{n!} \mathbb{E} g^{(n)}(X_{(...
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0
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Convex ordering of measures that are obtained by different push-forwards of a same measure
Suppose that we have a probability measure $\rho$ which is supported on $\mathbb{R}^d$ and absolutely continuous w.r.t. the Lebesgue measure. Take two vector fields $F, G : \mathbb{R}^d \rightarrow \...
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0
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Direct (first-order ?) algorithm to minimize $u(x) := \|x-a\|_C + r\|x\|_p$
Fix $a \in \mathbb R^n$, $r \ge 0$, $p \in \{1,2\}$, and a positive-definite matrix $C$ of order $n$. Define $u:\mathbb R^n \to \mathbb R$ by $u(x) := \|x-a\|_C + r\|x\|_p$, where $\|z\|_C := \sqrt{z^\...
- 6,726
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1
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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)$$
where $g$ is differentiable and convex, and $h$ are convex but not necessarily differentiable. If $g$ is the mean squared ...
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4
votes
2
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575
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Unit ball of the sum space
Let $V$ be a vector space and $\|\cdot \|_1$ and $\|\cdot\|_2$ two norms on $V$.
Let $\|\cdot\|_+$ be given by
$$ \|v\|_+ := \inf_{v = v_1 + v_2} \|v_1\|_1 + \|v_2\|_2 $$
It is well-known that $\|\...
- 37.6k
5
votes
3
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503
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How to prove this (corollary of) hyperplane separation theorem?
$X$ is a nonempty convex subset of $\mathbb{R}^n$ whose element is $x=\left(x_1,...,x_n\right)$.
The theorem is as follows.
If for each $x\in X$, there is an $i \in \left\{1,...,n\right\}$ such that $...
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votes
2
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454
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Proving the set $\left\lbrace \frac{(x + y)^2}{\sqrt{y}} \leq x - y + 5, y > 0 \right\rbrace$ is convex
I have recently picked up a course on Convex Analysis in my spare time, but feel I'm not quite up to speed with the 'tricks' for proving a set is convex.
I have managed to prove this by moving all ...
0
votes
2
answers
514
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Smooth approximation for non differentiable function
Let $f(t) = \min(\frac{1}{\lvert t\rvert}, 1)$. I would like to find a smooth approximating function $g$ such that $f(t) \leq g(t)$ for all real $t$. Is there a nice function $g$ out there? Any ...
- 3,595
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2
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227
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Hausdorff dimension of the non-differentiability set a convex function
Let $X \subset \mathbb R^d$ be open, $f : X \to \mathbb R$ and
$$
E := \{x \in X : f \text{ is not Fréchet differentiable at }x\}.
$$
Then we have the following result which is
Theorem: If $X= \...
- 851
-4
votes
1
answer
92
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Convex combination of $\frac{1}{x}$ inequality [closed]
Let $0 < x_1 \leq ... \leq x_n$ and $\sum \alpha_i = 1, \alpha_i \geq 0$. Show
$\sum \frac{\alpha_i}{x_i} \leq \frac{x_1 + x_n - \sum \alpha_i x_i}{x_1 x_n} $. Since the left side looks like a ...
3
votes
1
answer
336
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Derivative of distance function to a convex set in CAT(0) space
Let $(X,d)$ be a complete CAT(0) space. We denote by $T_x X$ the tangent cone at a point $x\in X$ and by $d_x$ its associated distance. So $(T_x X,d_x)$ is also a complete CAT(0) space. In CAT(0) ...
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0
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367
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Generalized Jensen's inequality for positively homogeneous functions
The function $f:V \to \hat{\mathbb{R}}$ is said to be positively homogeneous iff $f(\alpha v) = \alpha f(v)$ for every $\alpha \in \mathbb{R}_{++}$. Here $V$ is a real vector space and $\hat{\mathbb{R}...
- 509
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votes
1
answer
288
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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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