The convexity tag has no wiki summary.

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318 views

### Convexity of $\frac{1}{2}\vert x_1 + x_2 + x_3 - x_1x_2x_3\vert^2$

Anybody have any tips on how to show that the function $\frac{1}{2}\vert x_1 + x_2 + x_3 - x_1x_2x_3\vert^2$ is convex in $\mathbf{x}$, where $\vert x_i \vert \leq 1$?
This comes from the following ...

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**0**answers

228 views

### Are affine continuous functions on Bauer sub-simplices of the probability measures given by integration over continuous functions?

Let $X$ be a compact (non-metrizable) Hausdorff space and $\mathcal{P}(X)$ the set of Radon probability measures with
weak-$*$ topology (weak topology induced by the continuous functions).
Consider a ...

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votes

**1**answer

234 views

### Integral in a σ−convex set.

Having had no (proper) answer to this question, I formulate the remaining case as a new question as follows. With $I=[0,1]$, let $E$ be a separable (real) Banach space, and let $\gamma:I\to E$ be ...

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**2**answers

477 views

### A lower bound of a particular convex function

Hello,
I suspect this reduces to a homework problem, but I've been a bit hung up on it for the last few hours. I'm trying to minimize the (convex) function $f(x) = 1/x + ax + bx^2$ , where ...

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**3**answers

442 views

### surfaces of constant centro-affine curvature

It is well-known that every closed surface in $\mathbb R^3$ having constant Gauss curvature is a round sphere. Inspired by this question, I'd like to ask whether a similar rigidity holds for ...

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127 views

### Convexity of a cone in CxC

Hi. I have trouble deciding if the set of couples $ \left( \xi, \zeta \right) \in \mathbb{C}^2 $ with $ Re \left( \xi \text{ } \overline{\zeta} \right) > 0 $ is convex. It is a (real) cone, but is ...

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votes

**1**answer

554 views

### volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?

We denote by $\otimes_{\epsilon}$ the injective Banach tensor product.
Which is the asymptotic volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?

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votes

**2**answers

430 views

### Existence of extreme points

Let $(X,d)$ be a complete metric linear space whose balls are convex. Let $Y\subseteq X$ be a bounded, closed and convex subset that verifies the following property: for all $y_0\in Y$, the distance ...

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votes

**4**answers

2k views

### Polynomial roots and convexity

A couple of years ago, I came up with the following question, to which I have no
answer to this day. I have asked a few people about this, most of my teachers and some
friends, but noone had ever ...

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vote

**2**answers

481 views

### Convex polynomial homogenization and convexity

I have a polynomial that I know to be convex. If I homogenize the polynomial, is the resulting homogeneous polynomial also convex? I know that the perspective of a convex function is convex, but ...

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**0**answers

216 views

### Convexity and line segments

Let S be a subset of a linear space. Let S1 be the union of all line segments that join pairs of points in S. Now what happens if we repeat this process and construct S2, S3,....(Thus for example S2 ...

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**1**answer

354 views

### Approximation of an integral of a concave function

I suspect this is a homework question somewhere, but I've not seen it elsewhere and it seems like it should be easy: let $f(x)$ be a concave function from $[0,1]$ to the reals such that $f(0) = f(1) ...

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**0**answers

1k views

### Infinite convex combinations in a Banach space

Let's say that a subset $C$ of a Banach space $X$ is $\sigma$-convex if the following property holds:
For any sequence $(x_k)_{k\ge0}$ in $C$, and for
any sequence of non-negative real numbers ...

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**0**answers

902 views

### Two-convexity ⇒ Lefschetz?

Assume that
$\Omega$ is an open simply connected set in $\mathbb R^n$
(two-convexity) if 3 faces of a 3-simplex belong to $\Omega$ then whole simplex in $\Omega$.
Is it true that any component of ...

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votes

**2**answers

372 views

### Submanifolds lying on the boundary of a convex domain

Let $M$ be a submanifold of $\mathbb R^n$. Call $M$ locally convex if locally $M$ is contained in the boundary of a convex domain of $\mathbb R^n$.
Is there any known condition that is equivalent to ...

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**0**answers

174 views

### Radon transform and Log-concavity

This question is related to (but different from) that of Darsh Ranjan.
Is there a characterization of the functions $f:\mathbb R^n\rightarrow\mathbb R_{\ge0}$ whose Radon transform $\hat ...

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**0**answers

340 views

### Bounding point-wise maximum of the absolute difference of two convex functions

Let $\Delta: R \times R \rightarrow R_{+}$ be a positive and convex function (convex in, say, both the arguments) called the loss function.
Let $x \in R^d$. Moreover, let $H_1,...,H_r$ be sets of ...

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**1**answer

729 views

### If $f:R^n \to R$ is a smooth real-valued function such that $\nabla f : R^n \to R^n$ is a diffeomorphism, what can one conclude about the behavior of $f(x)$ at infinity?

This question may seem peculiar, so let me preface it by saying that it arose while I was trying to understand Legendre transformations better, and in that context it is fairly natural. Anyway, ...

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**1**answer

558 views

### Does a log-concave function on a convex set extend continuously to the boundary?

Let $U$ be an open convex set in a locally convex space $X$, and let $f : U \to [0,1]$ be a log-concave function on $U$ (i.e., bounded and real-valued). Under what conditions does $f$ have a ...

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359 views

### Does the dual Banach space $B(\ell^\infty)$ has weak* normal structure?

Let $K$ be a bounded closed convex subset of a Banach space $E$. A point $x$ in $K$ is called a diametral point if
$$
\sup_{y\in K} ||x-y||=diam(K).
$$
where $diam(K)$ denotes the diameter of $K$.
...

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votes

**2**answers

325 views

### If $K$ and $L$ are compact convex sets with smooth boundary, does their union have piecewise-smooth boundary?

Clarification: by "piecewise", I mean a finite number of pieces.
I'm sure this must be true, but my search for a citation was in vain (although I did learn the new term "polyconvex").
Thanks!

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**0**answers

127 views

### Lower semicontinuity of Bregman distances/divergences

For a Banach space $X$ and a convex functional $J:X \to [0,\infty]$ (i.e. with values in the extended reals), consider the associated Bregman distance: For $x,y\in X$ and $\xi\in\partial J(y)$:
...

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**1**answer

647 views

### Lipschitz properties of minima/minimizers of convex functions of two variables

Suppose I have a function $f(x,y)$ from $\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ that is convex in both $x$ and $y$. Set
$g(y) = \min_{x} f(x,y)$
What I would like is for $g(y)$ to be ...

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votes

**3**answers

570 views

### Some questions about Invexity

Recently, I am looking into a non-convex optimization problem whose points satisfying KKT conditions can be obtained. Then the problem becomes how to decide whether the KKT conditions are sufficient ...

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votes

**1**answer

269 views

### Can subgradient infer convexity?

It is known that
If $f:U→ R$ is a real-valued convex function defined on a convex open set in the Euclidean space $R^n$, a vector v in that space is called a subgradient at a point $x_0$ in $U$ if for ...

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votes

**1**answer

390 views

### Polygons and mirrors

Given a point $A$ inside a non-convex polygon $P$, is it always possible to place a finite set of mirrors (not necessarily along the boundary of $P$, any position inside $P$ is allowed) given by ...

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**1**answer

284 views

### Convex sets and projections

Hello!
I recently started (it's purely self-education) reading a "Mathematical programming and optimizations" book, did a vast part of the exercises related to the theoretical part and at one moment ...

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votes

**1**answer

565 views

### Orthogonal similarity of matrices

Given $M\in M_n({\mathbb R})$
and $\ell\in{0,\ldots,n-1}$, we define
$$d_\ell(M)=\sum_{j=1}^nm_{j,j+\ell},$$
where the indices are understood mod $n$. In particular, $d_0$ is the trace operator.
Let ...

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votes

**3**answers

522 views

### a different algebra/representation for convex sets

Hi,
I was dealing with finding a feasible region for a set of norm inequalities and the feasible region is convex. The question is not about how to find the feasible region but how to represent the ...

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votes

**1**answer

953 views

### Banach-Mazur distance between $\ell^p$-norms

Let $E^n$ be the real or complex space of dimension $n$. If $N$ and $M$ are two norms over $E^n$, and if $A$ is an endomorphism, then
$$\|A\|^M_N:=\sup_{x\ne0}\frac{M(Ax)}{N(x)}$$
is an operator norm ...

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**2**answers

796 views

### convex hull of k random points

suppose we have $k$ points placed uniformly at random in the unit cube in $\mathbb{R}^n$.
what is the probability that their convex hull has all of the $k$ points as extreme points?
[if it would be ...

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votes

**1**answer

382 views

### Convexity and Strong convexity of subsets of Surfaces

In the book Riemannian geometry - modern introduction by Isaac Chavel, three different definitions of convexity are introduced. I am looking for an example of a set which is convex but not strongly ...

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votes

**1**answer

245 views

### A differential inclusion relating to the slope of a convex function

This question is concerned with a possible lemma which would be very useful in one of my current research projects, but which I am currently unable to prove. The project as a whole relates to the ...

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**5**answers

10k views

### Is all non-convex optimization heuristic?

Convex Optimization is a mathematically rigorous and well-studied field. In linear programming a whole host of tractable methods give your global optimums in lightning fast times. Quadratic ...

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**9**answers

3k views

### Ways to prove an inequality

It seems that there are three basic ways to prove an inequality eg $x>0$.
Show that x is a sum of squares.
Use an entropy argument. (Entropy always increases)
Convexity.
Are there other means?
...

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**0**answers

3k views

### Volumes of Sets of Constant Width in High Dimensions

Background
The n dimensional Euclidean ball of radius 1/2 has width 1 in every direction. Namely, when you consider a pair of parallel tangent hyperplanes in any direction the distance between them ...

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729 views

### A characterization of convexity

While doing some research on polytopes I came to the following question. Maybe it's already somewhere but anyway I'll post it here.
Let $X\subset \mathbb{R}^3$ be such that, for every plane $P$, ...

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votes

**3**answers

2k views

### Covering a circle with red and blue arcs

We have a circle and two families of $n$ red arcs and $n$ blue arcs, positioned on the circle so that every two arcs of different colors intersect. Can one show that there is a point in the perimeter ...

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votes

**1**answer

469 views

### A variation on “Hearing the shape of a drum” for polytopes.

Let $\varphi:\mathcal S^{d-1}\longrightarrow \mathbb R_{>0}$ be a strictly positive function describing the boundary $\varphi(\mathbf x)\mathbf x,\mathbf x\in\mathbb S^{d-1}$ of a $d-$dimensional ...

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votes

**2**answers

842 views

### Proof that domains of positivity of symmetric nondegenerate bilinear forms are self-dual cones?

Max Koecher (in, for example, The Minnesota Notes on Jordan Algebras and Their Applications (new edition: Springer Lecture Notes in Mathematics number 1710, 1999)), defined a domain of positivity for ...

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vote

**6**answers

3k views

### Extreme point compact convex set.

The Krein-Milman theorem shows that a compact convex set in a Hausdorff locally convex topological vector space is the convex hull of its extreme points.
It seems this implies that a compact convex ...

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votes

**2**answers

689 views

### Maximal Ellipsoid

John's Theorem can be stated as "To every compact, convex body, there is a unique inscribed ellipsoid, whose volume is maximal among all inscribed ellipsoids." It goes on to classify this maximal ...

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**1**answer

405 views

### Determination of a symmetric convex region by parallel sections

This question is partly inspired by a problem in Stewart's Calculus: "Find the area of the region enclosed by $y=x^2$ and $x=y^2$."
Suppose $f\colon [0,1]\to [0,1]$ is a convex increasing function ...

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votes

**2**answers

276 views

### Efficiently sampling points from an integer lattice.

Let $\mathcal{L}$ = {x$\in$ $N^n$ : ||x||$_1$ $\leq$ m} denote the set of integer points in the positive orthant of the $\ell_1$ ball of radius $m$, where $m < n$. For each $x \in \mathcal{L}$, let ...

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votes

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2k views

### Compact Convex sets and Extreme Points

There are examples that show the set of extreme points of a compact convex subset of a locally convex topological vector space need not be closed when the real dimension of the space is at least 3. ...

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2k views

### If a quadratic form is positive definite on a convex set, is it convex on that set?

Consider a real symmetric matrix $A\in\mathbb{R}^{n \times n}$. The associated quadratic form $x^T A x$ is a convex function on all of $\mathbb{R}^n$ iff $A$ is positive semidefinite, i.e., if $x^T A ...

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votes

**4**answers

515 views

### Radstrom cancellation only for two convex sets?

I've seen this statement of Radstrom cancellation:
if $A +C \subset B+C$ where $A,B$ are convex, $B$ is closed, and $C$ is bounded, then $A \subset B.$
Is it essential that $A$ be convex?

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**12**answers

3k views

### Generalizations of the Birkhoff-von Neumann Theorem

The famous Birkhoff-von Neumann theorem asserts that every doubly stochastic matrix can be written as a convex combination of permutation matrices.
The question is to point out different ...

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**4**answers

1k views

### Minkowski sum of small connected sets

Suppose that the convex hull of the Minkowski sum of several compact connected sets in $\mathbb R^d$ contains the unit ball centered at the origin and the diameter of each set is less than $\delta$. ...

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votes

**3**answers

600 views

### A “round” lattice with low kissing number?

Historically, the lattices with high density were studied intensively, e.g. E_8 lattice or Leech Lattice. However, there are situations that lattices with low kissing number are required. ...