3
votes
1answer
145 views

Generalization of notion of convexity

I am searching for the correct term for the following, if it exists. A set $X\subset \mathbb{R}^2$ is called $r$-convex if for any two points $x_1, x_2\in X$ such that there exists an arc of radius ...
11
votes
2answers
286 views

Triangle with largest perimeter in a convex region

What is the largest value of $r$ such that the following statement is always true? "Let $C$ be a convex region with area $1$. There must exist a triangle contained in $C$ whose perimeter is at least ...
3
votes
1answer
74 views

Generalizations of Directly Similar Theorem?

There is an attractive theorem that says that if two plane figures are directly similar, then so is any convex combination of them. Below, $P_1$ and $P_2$ are directly similar polygons: they have the ...
5
votes
1answer
140 views

Shortest curve with given convex hull

Suppose $S\subset\mathbb{R}^2$ is compact and convex. Suppose $\Gamma:[0,1]\to S$ is a continuous curve that passes through every extreme point of $S$, i.e., the convex hull of $\Gamma([0,1])$ is $S$. ...
3
votes
1answer
185 views

Map from a convex polygon that increases distance

At the risk of asking an extremely stupid question, suppose that $P\subset\mathbb{R}^2$ is a convex polygon with area $1$ that contains the origin, and let $r$ denote the farthest distance between the ...
1
vote
4answers
2k views

How to find overlap between two convex hulls,along with the overlap area

I have two boundaries of two planar polygons , say,B1 and B2 of polygons P1 and P2(with m and n points in Boundaries B1 and B2). I want to find out if the Polygons overlap or not.If they overlap,then ...
6
votes
2answers
230 views

Diameter-area ratio for affine tranformations.

Consider an convex plane figure $F$. How to prove that there is an affine transformation $a$ such that $\sqrt{3}$ diameter$(a(F))^2\leq 4$ area$(a(F))$? I found only one reference, to "Über einige ...
35
votes
4answers
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 ...