fa.functional analysis – “Mollification” of a Convex Function with a Finite Set of Points Unchanged

$f:mathbb Rtomathbb R$ is a convex continuous function. We have a finite or a countable set of triples: ${(x_n,f(x_n),D_n)}_{nin N}$, where $D_n$ is the slope of one of the tangent line $L_n$ at $x_n$.

(If at a point $f$ is not differentiable, then multiple lines can be the tangent; $L_n$ is just one of those lines.)

Assuming that, for any $n,m,k$, the intersection of $L_n$ and $L_m$ cannot be the point $(x_k, f(x_k))$, then, we want to prove that, there exists a smooth function $g$, such that $g(x_n)=f(x_n)$ and $g'(x_n)=D_n$ for any $n$.


The original problem that I am trying to solve involves multi-dimensional manifolds but I think it is easy to generalize the 2-dimensional case.

By the mollification theorem a smooth function approximating $f$ must exists, but can it contain a set of points that was precisely the points on the graph of $f$?

real analysis – The class of function $C={f-gbig{|}f,g:mathbb{R}tomathbb{R}$ are convex$}$ is dense in the function space of $C(mathbb{R})

The class of function $C={f-gbig{|}f,g:mathbb{R}tomathbb{R}$ are convex$}$ is dense in the function space of $C(mathbb{R})={$all continuous functions between $mathbb{R}$ and $mathbb{R}}$

My idea: My teacher gave us a hint that we first need to prove that $C$ is closed under the lattice operation $vee$. This can be achieved if we note the fact that $(a_1-b_1)vee(a_2-b_2)=(a_1+b_2)vee(a_2+b_1)-(b_1+b_2)$. Then I do think the Baire category theorem should be applied because this theorem is essential when proving another similar proposition, that is, the class of continuous functions which are nowhere differentiable is dense in the function space of $C(mathbb{R})$. Now, what we have to do is to construct a series of nowhere dense closed sets $E_n$, whose union $bigcup_{ngeq1}E_n=C(mathbb{R})backslash C$. And what confuses me is how to use the hint given by my teacher to construct $E_n$.

Any hint or solution is highly appreciated!

general topology – Supporting set in convex sets

I am traing to prove that: If $f$ is a continuous linear functional on a compact convex set $C$ then the set $S={c in C : f(c)=maxf(C)}$ is non empty convex and supporting set in $C$.where $C$ is convex set.

I’m using this definition of supporting set : $S$ is
called a supporting set in $C$ if S is convex and has the property that if an “inside
point” of a line segment from $C$ belongs to $S$, then the entire line segment must
belong to $S$.That is, if $c_{1},c_{2} in C$ and $lambda c_{1} +(1- lambda)c_{2} in S $ for some $0 < lambda < 1$, then both $c_{1}$ and $c_{2}$ must be in $S$.

Any idea?

unity – Why use Convex Polygons and not Concave ones in path-finding?

I read in Unity’s path-finding documentation that they use convex polygons because there won’t be any ‘obstruction’ between 2 points. Then they add their vertices as nodes along with starting and ending points and traverse them using A* algorithm to reach the required destination.

However, I do not understand what they mean by “no obstruction between 2 points”. I tried to check the differences between concave and convex polygons but only the angle differences come up (in convex the interior angles must be less than 180 degrees)

If a minimization task is a convex optimization problem, is its maximization also?

If a minimization task is a convex optimization problem, is the maximization of the same objective function also always a convex optimization problem?

My guess is yes since minimization of the negative of the objective function is maximization, but wondering if there are cases that outdo, and disprove, this sign ‘trick’

linear algebra – Bidual of a convex cone is its closure in finite dimensional complex space $V$

Let $S subset V$ where $V$ is a finite dimensional complex space. Let dual of $S$ be $f in V^*$ s.t. $f(S) ge 0$ and bidual of $S$ be $S^{**}$. The question is whether the bidual of a convex cone is its closure in finite dimensional complex space $V$. This is obviously true in finite dimensional real space.

Let $S$ be a convex cone, i.e. $S$ is convex and $tS subset S$ for $tge 0$. View $V$ as real space we can find a real linear functional $Lambda$ for each $p notin bar{S}$ s.t. $Lambda(bar{S}) ge 0$ but $Lambda(p) < 0$.

discrete geometry – Source request: Optimal bounds on signings of points from a convex body

I recently came across an old survey of problems in discrete geometry: https://pdfs.semanticscholar.org/c350/f4d4a9466fa6708d99ec1187c63d89bed20f.pdf

Problem 2.1 from the list caught my eye. It states the following (more or less):

Let $y_1, y_2, … y_m in mathbb{R}^k$, $y_i neq 0$. Let $K$ be the convex body consisting of all points $z = sum_{i=1}^m c_i y_i$ for $c_i in (-1, 1)$. Does there exists a universal constant $C$ such that given $x_1, x_2, … x_k in K$, there are signs $delta_1, delta_2, … delta_k in {-1,1}$ such that $sum_{i=1}^k delta_i x_i in C sqrt{k} K$?

Has any progress been made on this problem? I have not been able to find anything in the literature. Further, Schectman claims that a bound of $C sqrt{k log log k}$ is possible: I have not been able to find this in the literature. Does anyone know a source for this claim?

fa.functional analysis – Closed convex hull in infinite dimensions vs. continuous convex combinations

No. Even in one dimension. Say $K$ is the open interval $(0,1)$. Show $0 notin K^*$. Let $mu$ be a probability measure with support contained in $(0,1)$. Indeed,
$$
r(mu) := int_K x,dmu(x)
$$

is the integral of a positive function. That is, $x > 0$ a.e. So $int_K x,dmu(x) > 0$. Similarly $1 notin K^*$.

In a Banach space $E$, if there is any extreme point of $M = overline{text{conv} K}$ that does not already belong to $K$, then it also does not belong to $K^*$. So what if $K$ is the set $text{ex}; M$ of extreme points of $M$? Can we recover $M$ as $K^*$?

A very nice little book that discusses this situation is

Phelps, Robert R., Lectures on Choquet’s theorem, Lecture Notes in Mathematics. 1757. Berlin: Springer. 124 p. (2001). ZBL0997.46005.

Choquet’s theorem tells us that every point of a compact convex set $M$ is of the form $r(mu)$ for some probability measure concentrated on the set $text{ex}; M$ of extreme points of $M$.


Plug
My first publication to attract any notice was this one, where there is a generalization of Choquet’s theorem to certain closed bounded noncompact sets $M$.

Edgar, G. A., A noncompact Choquet theorem, Proc. Am. Math. Soc. 49, 354-358 (1975). ZBL0273.46012.