Convex analysis is the study of properties of convex sets and convex functions. For questions about optimization of convex functions over convex sets, please use the (convex-optimization) tag.

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Convexity of $\mathrm{trace}(S) + m^2\mathrm{trace}(S^{-2})$

I have the following function $f(S)=\mathrm{trace}(S)+m^2\mathrm{trace}(S^{-2})$ where $S\in \mathcal{M}_{m,m}$ symmetric positive definite matrix. I'm trying to prove the convexity of this function ...
10
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2answers
251 views

Convexity of a complicated function

Let $\mathbb{S}$ be a 2-D convex set in the positive quadrant. Let us define \begin{align} y_L=\min_{(x,y)\in\mathbb{S} }y \\ y_R=\max_{(x,y)\in\mathbb{S} }y \end{align} For any positive number ...
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4answers
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Why does a convex set have the same interior points as its closure?

Let $C$ be a convex subset of $\mathbb{R}^n$. I've been trying for hours to prove that $\dot{\overline{C}}=\dot{C}$. Somehow my intuition completely fails me. I found a proof in a textbook, but just ...
6
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1answer
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Is the trace of inverse matrix convex?

Hi I would like to know whether the trace of the inverse of a symmetric positive definite matrix $\mathrm{trace}(S^{-1})$ is convex. Actually I know that the trace of a symmetric positive definite ...
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2answers
500 views

How to show that $\mathbb R^n$ with the $1$-norm is not isometric to $\mathbb R^n$ with the infinity norm for $n>2$?

Could you please give me a hint to prove that $\mathbb{R}^n$ with the 1-norm $\lvert x\rvert_1=\lvert x_1\rvert+\cdots+\lvert x_n\rvert$ is not isometric to $\mathbb{R}^n$ with the infinity-norm ...
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2answers
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Dual of a dual cone

Any hint on how to prove the following please: Let $K$ be a convex cone, and $K^*$ its dual cone. Prove that $K^{**}$ is the closure of $K$. Thanks!
4
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1answer
464 views

Do projections onto convex sets always decrease distances?

Suppose $(M, d)$ is some $\ell_p$ metric space (not necessarily Euclidean), and $C \subseteq M$ is a closed convex set. Consider the projection function $f_C:M\rightarrow C$ defined such that: ...
2
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1answer
94 views

minimizing $\sum_{i=1}^n \max(|x_i - x|, |y_i - y|)$

Suppose there are $n$ points $(x_i, y_i)$ for $i = 1,\ldots,n$. Please find another point $(x, y)$ to minimize function: $$\sum_{i=1}^n \max(|x_i - x|, |y_i - y|)$$
2
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1answer
470 views

Prove that the convex hull of a set is the smallest convex set containing that set

How do you prove that the convex hull of A is the smallest convex set containing A? edit: definition of a convex hull: Given a set A ⊆ ℝn the set of all convex combinations of points from A is called ...
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1answer
133 views

interior points and convexity

Question: Let If $P\subseteq \mathbb{R}^n$ be a convex set. show that $int(P)$ is a convex set. I know that a point $x$ is said to be the interior point of the set $P$ if there is an open ball ...
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can any continuous function be represented as a sum of convex and concave function?

I read that any continuous function can be represented as a sum of convex and concave function, meaning for all $f(x)$, $f(x) = g(x) + h(x)$ where $g$ is convex and $h$ is concave. There could be ...
17
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1answer
490 views

Is $[0,1]^\omega$ homeomorphic to $D^\omega$?

Let $n\in \mathbb N$ and let $D^n$ be the closed $1$-ball in $\left(\mathbb R^n, \|\,\cdot\,\|_1\right)$. It is not too hard to show that $[0,1]^n \cong D^n$ in this case. This observation leads to ...
8
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1answer
721 views

A real differentiable function is convex if and only if its derivative is monotonically increasing

I'm working on a problem in baby Rudin, Chapter 5 Exercise 14 reads: Let $f$ be a differentiable real function defined in $(a,b)$. Prove that $f$ is convex if and only if $f'$ is ...
14
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1answer
264 views

Expected size of subset forming convex polygon.

If there are $4$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the interval $\left(0,1\right)$, what is the expected largest size (or ...
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2answers
2k views

Is the composition of $n$ convex functions itself a convex function?

Is a set of convex functions closed under composition? I don't necessarily need a proof, but a reference would be greatly appreciated.
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2answers
432 views

Convex combination in compact convex sets.

Let's $K\subset\mathbb{R}^n$ compact. If $K$ is convex then who to prove that any point of $K$ is convex combination of one or two extremal points of $K$? Intuitively, for any closed ball that is ...
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1answer
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Convexity of the product of two functions in higher dimensions

Exercise 3.32 page 119 of Convex Optimization is concerned with the proof that if $f:\mathbb{R}\rightarrow\mathbb{R}:x\mapsto f(x)$ and $g:\mathbb{R}\rightarrow\mathbb{R}:x\mapsto g(x)$ are both ...
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3answers
188 views

How to prove that $e^x$ is convex? [closed]

I need a help with proving of $e^x$ function. We know $e^x$ is a convex function. $$f(c(x')+(1-c)(x'')) ≤ c\cdot f(x')+(1-c)\cdot f(x''), \quad 0 ≤ c ≤ 1$$ Thanks.
2
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2answers
106 views

When does it make sense to define a generator of a set system?

In a set system, such as a topology, sigma algebra or convexity structure, a generator is defined to be a class of subsets such that the given set system is the coarsest such set system ...
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2answers
180 views

Is this set compact?

Given two densities $f$ and $g$ the squared Hellinger distance between them is defined as follows $$ S(g,f)=H^2(g,f)=\frac{1}{2}\int_{\mathbb{R}}\left(\sqrt{g(y)}-\sqrt{f(y)}\right)^2 \mbox{d}y $$ I ...
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0answers
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can we prove that a certain supremum of affine functions is frechet differentiable or at least continuous?

Let $X$ be a Hilbert space. Let $A\colon \operatorname{dom} A\to X$ be linear operator with closed graph (not necessarily bounded). Define $$ g\colon \operatorname{dom}A\to \mathbb{R}:x\mapsto ...
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1answer
86 views

A weak version of Markov-Kakutani fixed point theorem

Let $\emptyset \not = X\subseteq \Bbb{R}^n$ be convex and compact and let $\cal{A}$ be a commuting family of affine maps from $\Bbb{R}^n$ into $\Bbb{R}^n$ such that $X$ is invariant under each element ...
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0answers
175 views

how to solve the following expectation? closed-form expression or approximation

Suppose there is a binomial random variable $X\sim B(n-1,p)$,how to solve the following expectation $$E[(1- b^{X})^{m}]$$ where $b\in (0,1]$ and $m\in \mathbb{N} $ are all constants.I have tried my ...
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1answer
211 views

Intersection of Two Simplices

How to find vertices a the polytope-intersection of two simplices, if I know the vertices of these simplices. More precisely: Let $T_1$ and $T_2$ be two regular $n-1$ dimensional simplices with ...
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7answers
2k views

Operations research book to start with

for somebody having a quite strong background in Mathematics, which are some good books for the domain of Operations research? I guess there are textbooks covering topics like linear and nonlinear ...
10
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1answer
485 views

Showing $\left|\frac{a+b}{2}\right|^p+\left|\frac{a-b}{2}\right|^p\leq\frac{1}{2}|a|^p+\frac{1}{2}|b|^p$

For $a,b \in \mathbb R$, $p\geq2$ I try to show $$\left|\frac{a+b}{2}\right|^p+\left|\frac{a-b}{2}\right|^p\leq\frac{1}{2}|a|^p+\frac{1}{2}|b|^p.$$ Is this a popular inequality (At least I could not ...
5
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1answer
210 views

Dual to the dual norm is the original norm (?)

I have the following questions about dual norms : How do you prove that the dual of the dual norm is in fact the original norm? This is what I have so far: If I have $\|y\|_* $ as the norm dual of ...
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3answers
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How to prove that if convex $A \subset \mathbb{R}^n$ and $A+A=A$ then $0 \in cl(A)$?

How to prove that if convex $A \subset \mathbb{R}^n$ and $A + A = A$ then $0 \in cl(A)$? For a example of $A$ which holds a condition but $0 \notin A$ consider $(0,\infty)$.
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1answer
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Positive definite Hessians from strictly convex functions

Let $f: D \to \mathbb{R}\ $ be a function on non-singular, convex domain $D \subseteq \mathbb{R}^d$ and let us assume the second-order derivatives of $f$ exist. It is well known that $f$ is convex if ...
8
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1answer
310 views

Sequence of convex functions

If a sequence of convex functions $\{ f_n \}$ on [0,1] converges pointwise to a continuous function f, then is the convergence uniform? The questions is almost identical to this one, except that the ...
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2answers
186 views

About the hyperplane conjecture.

I have recently heard about the hyperplane conjecture and I would like to understand better the problematic behind this conjecture. The hyperplane conjecture: There exists a universal constant ...
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0answers
189 views

How does one compute a bounding simplex for a set of points?

An algorithm I've implemented to tessellate an N-dimensional space requires starting with a bounding N-simplex. Given a set of $k$ points $S_{0..k-1} \subset R^N$ is there a procedure to find a ...
3
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2answers
141 views

Show that $(1+x)(1+y)(1+z)\ge 8(1-x)(1-y)(1-z)$.

If $x>0,y>0,z>0$ and $x+y+z=1$, prove that $$(1+x)(1+y)(1+z)\ge 8(1-x)(1-y)(1-z)$$ Trial: Here $$(1+x)(1+y)(1+z)\ge 8(1-x)(1-y)(1-z) \\ \implies (1+x)(1+y)(1+z)\ge 8(y+z)(x+z)(x+y)$$ I ...
2
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2answers
172 views

Optimize a log det function with respect to a matrix, and the saddle point analysis

Suppose I want to to find the local minima of a logdet function $\mathcal{L}$ with respect to a Matrix $\mathbf{A}$, $$ \mathcal{L} = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - ...
2
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2answers
266 views

fail of cantor intersection property on closed , bounded , convex sets of integrable functions

This is from my recent homework. I am asked to find a descending nested sequence of closed , bounded , nonempty convex sets $\{D_n\}$ in $L^1(\mathbb{R})$ such that the intersection is empty , where ...
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2answers
523 views

If the Minkowski sum of two convex closed sets is a Euclidean ball, then can the two sets be anything other than Euclidean balls?

If for two convex closed sets $S_1$ and $S_2$, the Minkowski sum is a Euclidean ball then can $S_1$ and $S_2$ be anything other than Euclidean balls themselves. I suspect they can be but I haven't ...
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1answer
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A bounded subset in $\mathbb R^2$ which is “nowhere convex”?

Let $F : \mathbb S^1 \to \mathbb R^2$ represents a simple closed curve $C$ in $\mathbb R^2$. The Jordan curve theorem says that the curves bounds a interior domain $\Omega$ and $\partial \Omega= C$. ...
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2answers
953 views

Is the convex hull of closed set in $R^{n}$ is closed?

Is convex hull of closed set in $R^{n}$ closed?
5
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1answer
213 views

When does it make sense to define a base of a set system?

In a topology, a base is defined to be a class of subsets such that every open set is the union of some members of it. In a convexity structure, a base is defined to be a class of subsets ...
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2answers
358 views

Weakly convex functions are convex

Let us consider a continuous function $f \colon \mathbb{R} \to \mathbb{R}$. Let us call $f$ weakly convex if $$ \int_{-\infty}^{+\infty}f(x)[\varphi(x+h)+\varphi(x-h)-2\varphi(x)]dx\geq 0 \tag{1} $$ ...
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1answer
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Cancellation law for Minkowski sums

Let $(X,\|\cdot\|)$ be a Banach space and $A,B,C\subset X$ closed bounded non-empty convex subsets. Let $+$ denote the Minkowski symbol for addition. Does the $+$ satisfy: $$A+C\subset B+C\implies ...
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3answers
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Prove one set is a convex hull of another set

Define two sets: $A = \{x \in \{0,1\}^n : \lVert x \rVert_1 \leq k\}$ is a finite set of binary vectors; $B = \{x \in [0,1]^n : \lVert x \rVert_1 \leq k\}$ is an infinite set of real-valued vectors, ...
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2answers
783 views

How to prove convexity?

Let us consider the function $$I(p):= \frac {\Gamma(2-p)\Gamma(3p)}{(p\Gamma(p))^2} $$ on the interval $(0,1),$ where $\Gamma(x)$ denotes the gamma function. How to prove its convexity there?
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2answers
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When finding root, does Newton's method fail if the function is non-differentiable?

According to wikipedia's description, the Newton's method finding a root presumes a differentiable function. Then, will it fail when encountering non-differentiable function? For example, can it find ...
3
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1answer
194 views

Effect the zero vector has on the dimension of affine hulls and linear hulls

I am currently working through "An Introduction to Convex Polytopes" by Arne Brondsted and there is a question in the exercises that I would like a hint, or a nudge in the right direction, please no ...
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1answer
50 views

Relation between mean width and diameter

Question: Let $A$ be a compact set in $\mathbb R^n$. Is it always true that $\text{mean-width}(A) \ge C \cdot \text{diam}(A)$ for some constant $C$ depending only on the dimension? If not, is it ...
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0answers
129 views

Infimum over area of certain convex polygons

Let us identify the Euclidean plane with the complex plane $\Bbb C$. For every $n\ge 1$, consider the following collection of finite sets in $\Bbb C^{n+1}$: $$S_n:=\{(z_0,z_1,\dots,z_n)\in\Bbb ...
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0answers
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Find a tight upper bound of the following expectation.

I am stuck in finding a tight upper bound (as tight as possible) of the following expectation $$E\left [ (1-a\cdot b^{X})^{m} \right ]$$ where $X\sim B(n-1,p)$ is a binomial random variable.In ...
2
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1answer
101 views

Mathematical expectation is inside convex hull of support

Let $\xi$ be a random variable supported in some set $A \in \mathbb{R}^n$: $\xi \in A$ a.e. How to show that $\mathsf E \xi \in \mathop{\mathrm{conv}}{A}$? Let $s(x)$ be a support function of set ...
2
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1answer
310 views

Derivative of Convex Functional

Suppose that $H$ is a real Hilbert space and that $f:H \to \mathbb{R}$ is differentiable in the Frechet sense. Then we can think of the derivative as a function $f': H \to H^* = H$. Suppose that this ...