# Tagged Questions

**2**

votes

**1**answer

102 views

### Quantitative stability: Hausdorff distance between subdifferentials

Suppose I have two convex functions $f$ and $g$ mapping $\mathbb{R}^{n}\to\mathbb{R}$ (so they are 'more' than proper). Suppose $\|f-g\|_{\infty}<\epsilon$, the sup-norm. In particular, the ...

**0**

votes

**0**answers

97 views

### Can we define log-convex operators?

Let $I\subset\mathbb{R}$. A function $f:I\rightarrow\mathbb{R}$, is said to be log-convex if $\log f$ is convex or equivalently for all $x,y\in I$ and $\alpha\in [0,1]$
$$f(\alpha x+(1-\alpha)y)\leq ...

**1**

vote

**0**answers

54 views

### How is this transformation related to the Legendre transform?

I stumbled over the following transform in a statistical mechanics paper:
Unfortunately, no mathematical details were given there, which is why I wanted to ask here about this transform.
Let $s : ...

**5**

votes

**1**answer

113 views

### Is this graph of reciprocal power means always convex?

Let
$$
p = (p_1, \ldots, p_n)
$$
be a finite probability distribution, which for convenience I'll assume to have no zeroes: thus, $p_i > 0$ for all $i$ and $\sum_i p_i = 1$.
Is the function
...

**5**

votes

**2**answers

134 views

### Smooth convex extensibility of combination of two line segments

This is a refined version of my earlier question Convex extensibility of combination of two lines.
Is there a smooth function $f:[0,1]\times [0,1]\rightarrow\mathbb R$ such
that for all $x\in ...

**3**

votes

**1**answer

207 views

### About generalized Minkowski inequality

For which functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ does the inequality
$f^{-1}\left(\sum\limits_{k=1}^n f(x_k+y_k)\right) \leq f^{-1}\left(\sum\limits_{k=1}^n f(x_k)\right) + ...

**1**

vote

**0**answers

92 views

### Linearization of cones

Suppose that $K$ is a closed convex cone in $R^{n}$. Is there a "nice" function $f:R^{n} \rightarrow R^{m}$ so that $f(K)$ is a subspace? What about an approximate subspace?

**1**

vote

**1**answer

225 views

### When does the finite union of convex sets have a hole in it?

Let $f_1, \dots, f_j$ be convex functions from $\mathbb{R}^n \to \mathbb{R}$. I am trying to develop a test that decides whether or not the set $\{x | f_1(x) \le k_1\} \cup \dots \cup \{x | f_n(x) ...

**3**

votes

**1**answer

278 views

### Possible to find a set of log-concave functions with log-concave sums?

While the set of log-convex functions is closed under addition, the set of log-concave functions is not. Yet if $f$ is log-concave, $\ln(k f) = \ln(k)+\ln(f)$, with $k \in \mathbb{R}^+$ constant, is ...

**1**

vote

**1**answer

134 views

### Smallest Lipschitz constant on non-convex domains

It is well known that if a function $f:U\to \mathbb C^n$, $U\subset \mathbb C^m$ satisfies $\sup_{x\in U}\|Df(x)\|_{\infty} = C < \infty$ uniformly on $U$ and $U$ is compact and convex, then $f$ is ...

**2**

votes

**3**answers

387 views

### A consequence of convexity

Let $f:\mathbb{R}\to\mathbb{R}$ a convex decreasing function. Let $x_0 < x_1 < x_2$.
Studying the behaviour of the difference quotient, it is clear that
$$f(x_0)-f(x_2) \leq M (f(x_0)-f(x_1))$$
...

**6**

votes

**0**answers

195 views

### Is $Q_n(x)=\sigma_{n+1}(x)/\sigma_n(x)$ logarithmically convex on $\mathbf{R}$?

In 1975 J. van de Lune considered the monotony properties of the canonical Riemann Upper and Lower sums for $\int_0^1 t^xdt$, with $x>0$.
Writing $\sigma_n(x) := 1^x+2^x+\cdots+n^x$ these sums are
...

**1**

vote

**1**answer

254 views

### Conjecture that two nested convex curves have a point with the same slope

I'm trying to prove a conjecture and need some help.
Consider a continuous, twice differentiable function $p(a)$ such that $p(0) = 0$ and $\forall a$, $p'(a) > 0$ and $p''(a) < 0$ and $p$ is ...

**1**

vote

**1**answer

530 views

### proving the interior of a dual cone is the set of vectors whose inner product is strictly positive on the cone

Apologies for posting such a simple question to mathoverflow. I've have been stuck trying to solve this problem for some time and have posted this same query to math.stackexchange (but have received ...

**14**

votes

**3**answers

598 views

### A natural center of a convex weakly compact set in Banach space

Question: Let $S$ be a convex weakly compact set in Banach space $H$. Propose a natural way to define the unique center $O \in S$.
Motivation: A lot! For example, in game theory $S$ can be a set of ...

**17**

votes

**1**answer

735 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, ...

**5**

votes

**1**answer

562 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 ...

**2**

votes

**1**answer

247 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 ...