All Questions
Tagged with plane-geometry convex-geometry
45 questions
45
votes
4
answers
5k
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 no one had ever ...
11
votes
5
answers
695
views
What is the area of the biggest open convex set inside the unit square not containing k points?
Given $k\in \mathbb N$, and $k$ points inside the unit square there should be an arrangement that minimizes the area of the biggest open convex set inside the unit square not containing these points.
...
11
votes
2
answers
1k
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 ...
- 111
10
votes
3
answers
537
views
Perimeter-halving center of a convex shape
Let $P$ be a convex polygon (or any convex body in $\mathbb{R}^2$)
with perimeter of length $1$. Call a chord $c$ of $P$ perimeter-halving
if half the perimeter lies to one side of $c$
(and so half to ...
- 151k
10
votes
0
answers
343
views
Bang's open question strengthening Tarski's planks problem
Tarski's Planks problem,
solved by Thøger Bang in 1951, says (in a simplified $\mathbb{R}^2$ version) that it requires
"planks" (parallel strips) of total width $\ge d$ in order to completely cover
a ...
- 151k
9
votes
3
answers
563
views
Visual proof of convergence for Steiner's symmetrization
I want to find a visual proof of the following fact:
For any convex figure in the plane there is a sequence of Steiner's symmetrizations that makes it arbitrary close to a circular disc.
All ...
- 45k
7
votes
1
answer
186
views
Decidability of convex rearrangements of polygons
Triggered by the MO question,
"How many convex shapes can be made with the pieces of the Stomachion?," I would like to pose this question:
Q. Given $n$ polygons in a set $S$, say each with integer ...
- 151k
7
votes
1
answer
412
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 $S$. ...
- 363
7
votes
1
answer
317
views
Minimum area of the convex hull of the union of a parallelogram and a triangle
This question is somewhat dual to my previously stated question about Maximum area of the intersection of a parallelogram and a triangle, where the triangle and parallelogram each is assumed to be of ...
- 7,276
6
votes
2
answers
459
views
A (possibly boring) Voronoi Game
The board for this game is a compact convex region $\cal C$ of $\mathbb{R}^2$.
Below I illustrate with $\cal C$ an equilateral triangle.
Two players, $A$ and $B$, alternate turns.
At each turn they ...
- 151k
6
votes
2
answers
207
views
On reflection properties of convex regions
It is well known that any ray of light passing thru a focus of an ellipse will pass thru the other focus after a single reflection from the ellipse boundary. If $A$ and $B$ are the foci of an ellipse, ...
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6
votes
2
answers
515
views
On 'fair bisectors' of planar convex regions
Definitions (https://www.ias.ac.in/article/fulltext/pmsc/122/03/0459-0467):
Given a planar convex region $C$ (could be smooth or polygonal), an area bisector of $C$ is any line that partitions $C$ ...
- 5,979
6
votes
1
answer
127
views
Convex planar regions with all area bisectors having equal length
Following A claim on the concurrency of area bisectors of planar convex regions, let me record a couple of simple queries.
An area bisector (perimeter bisector) of a planar convex region is a chord ...
- 5,979
5
votes
4
answers
24k
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, ...
- 53
5
votes
2
answers
368
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 ...
- 5,063
5
votes
0
answers
139
views
On convex regions containing (and contained within) a given triangle
Given an arbitrary triangle T.
How does one find the convex region C_M of largest area containing T such that T is also the largest area triangle that is contained within C_M?
Guess: for any T, ...
- 5,979
4
votes
1
answer
256
views
Polar interpretation of convexity
Let $C$ be a convex polygon in the plane containing the origin, and let $r(\theta)$ for $\theta\in[0,2\pi)$ be a parametrization of its boundary. Is there a condition on $r$ that is equivalent to (or ...
- 645
4
votes
1
answer
356
views
Left and right halves of convex curve
Let $S$ be a set of $n$ points in the plane in general position (no 3 on a line), $n$ even.
A halving line is a line through $2$ points of $S$ that partitions $S$ into 2 equal parts ($(n-2)/2$ points ...
- 214
4
votes
0
answers
90
views
recursively convex plane curves
For the lack of a better term let's call a convex simple loop $u(t)$ recursively convex if for any $n \geq 0$ the $n$-th derivative $u^{(n)}(t)$ is a convex simple loop. We conjecture that any ...
- 181
3
votes
1
answer
178
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 ...
- 151k
3
votes
1
answer
237
views
Find the number of triangles in plane
Let $S$ be a set of $n$ points in the plane in general position. Each 3 points of S span a triangle. Total number of triangles spanned by S:
$$\binom{n}{3}=\frac{n(n-1)(n-2)}{6}=\frac{1}{6} n^3-O(n^2 )...
- 214
3
votes
1
answer
190
views
On some centers of convex regions based on partitions
These questions are inspired by Yaglom and Boltyanskii's 'Convex figures'.
Winternitz Theorem: If a 2D convex figure is divided into 2 parts by a line $l$ that passes through its center of gravity, ...
- 5,979
3
votes
1
answer
247
views
Map from a convex polygon that increases distance
At the risk of asking an extremely stupid question, suppose that $P\subset\mathbb{R}^2$ is a convex polygon with area $1$ that contains the origin, and let $r$ denote the farthest distance between the ...
- 33
3
votes
1
answer
349
views
What is the shape of the convex $n$ -gon which gives the maximum of a function?
Supposing that the length of every edge of the convex $n$-gon $P_1P_2$$\cdots$$P_n$ is 1, what is the shape of the $n$-gon which gives the maximum of the following function $B_n$? $$B_n=\sum_{1\le{i}\...
- 4,757
3
votes
1
answer
288
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 $...
2
votes
1
answer
123
views
Segments on a closed convex plane curve
Is it true that there does not exist a closed convex plane curve containing an infinite number of segments, belonging to distinct lines each?
- 123
2
votes
1
answer
138
views
On the moment of inertia of planar convex regions and possible special nature of circular disks
We consider uniform convex planar regions and lines through their center of mass and lying in the same plane as the region; each line is parametrized by an angle $\alpha$ it makes with some reference ...
- 5,979
2
votes
1
answer
179
views
On segments of equal area cut from planar convex regions by chords
Consider a planar convex region $C$ of unit area and all chords of it that cut off a segment of area $\alpha$ from $C$. Obviously, if $C$ is a circular disk of unit area, all segments of area $\alpha$ ...
- 5,979
2
votes
1
answer
131
views
Curves of constant width that contain triangles
Wikipedia references: Curve of constant width,
Reuleaux polygon.
We record a pair of questions on the same lines as Smallest 3-ellipses that contain triangles.
Questions:
How does one find and ...
- 5,979
2
votes
1
answer
258
views
Smallest 3-ellipses that contain triangles
Reference: https://en.wikipedia.org/wiki/N-ellipse
Question: How does one find and characterize the smallest 3-ellipses (n-ellipses with n =3) that contain a given triangle? 'Smallest' can mean 'least ...
- 5,979
2
votes
0
answers
100
views
Rolling/width functions: Characterization?
Let $K$ be a strictly convex planar body of perimeter $1$.
Roll it along the $x$-axis from $0$ to $1$, and
define $f(x)$ to be the height of the highest point of $K$
when it touches at $(x,0)$.
So $f(...
- 151k
2
votes
0
answers
56
views
Projecting a convex partition onto a convex set
Say that $X$ and $Y$ are two convex regions in the plane, and suppose that $X \subset Y$. Further suppose that $Y$ is partitioned into disjoint convex subsets $Y_1 ,\dots, Y_n$. Is there a way of ...
- 4,049
2
votes
0
answers
126
views
Intersecting balls with convex regions and a bisector thereof
This question is related to my previous posting
Angle subtended by the shortest segment that bisects the area of a convex polygon
Let $C$ be a convex region in the plane and let $s$ be the shortest ...
- 113
1
vote
2
answers
157
views
A claim on the concurrency of area bisectors of planar convex regions
We add a little bit to On 'fair bisectors' of planar convex regions and Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia
Definitions: Given a ...
- 5,979
1
vote
2
answers
153
views
Smallest triangles that contain 2D convex regions with reflection symmetry
Given any 2D convex region $C$ with a mirror symmetry. Two pairs of questions:
We need to find the smallest area (likewise, smallest perimeter) triangle that contains $C$. Is it sufficient to only ...
- 5,979
1
vote
1
answer
82
views
Enclosing a convex plane domain in a disc
The following statement seems obvious to me:
Let $\gamma:S^1\to\mathbb R^2$ be a smooth injection such that $\dot\gamma$ and $\ddot\gamma$ never vanish.
Then $\gamma$ encloses a strictly convex ...
- 8,086
1
vote
1
answer
267
views
Vertices of Curves and Eigenvectors of Hessian
This might be a trivial question, but I can't seem to figure it out. Suppose I have an implicitly defined curve in the plane given by $f(x,y) = t$.
This curve is strictly convex, and feel free to ...
- 115
1
vote
0
answers
49
views
On points in the interior of planar convex regions and inscribed triangles
Given any planar convex region C, it is easy to show that every point in the interior C is the mid point of at least one chord of C. Likewise,
Question: Is every point in the interior of C the ...
- 5,979
1
vote
0
answers
65
views
Minimum bounding rectangle of symmetric convex bodies in the plane : is the ball the worst case
The minimal rectangle containing the euclidean ball in the plane is the standard cube $B_\infty = [-1;1]^2$. I would like to know if the euclidean ball is the worst symmetric convex body to be ...
- 245
1
vote
0
answers
65
views
To extend the Steiner-Lehmus theorem
The Steiner Lehmus theorem (https://en.wikipedia.org/wiki/Steiner%E2%80%93Lehmus_theorem) states: Every triangle with two angle bisectors of equal lengths is isosceles.
Question: What could one say ...
- 5,979
1
vote
1
answer
227
views
On comparing planar convex regions of equal perimeter and area
Definitions:
The Hausdorff distance between two point sets is the greatest of all the distances from a point in one set to the closest point in the other set.
Given two planar convex regions $C_1$ ...
- 5,979
0
votes
2
answers
3k
views
Determine the boundary points of a set of points [closed]
I have a set of points $S=\{(x_1,y_1),(x_2,y_2),\ldots,(x_n,y_n)\}$. Then how to find the boundary points (which is a subset of $S$) of $S$?
There are methods like convex hull, concave hull and $\...
- 17
0
votes
1
answer
126
views
On 'special' points on uniform planar convex regions defined in terms of moment of inertia
The following can be easily proved using perpendicular axes theorem and intermediate value theorem:
Lemma: Given any uniform planar convex region $C$ and any point on it, $P$, there will be at least ...
- 5,979
0
votes
1
answer
55
views
On 'axiality' of planar convex regions
Axiality has been studied under a definition given here: https://en.wikipedia.org/wiki/Axiality_(geometry)
Consider an alternative definition of axiality as follows: For a convex region C, consider a ...
- 5,979
0
votes
0
answers
36
views
What is the locus defined by those equations?
I would like to know what is the locus of $x \in \Bbb R_+^n$ ($n=2$ would already be fine) defined by
$\sum a_i \cdot x_i$ s.t. $a_i+\epsilon \geq 0$, $\epsilon \in \Bbb R$.
I know that if $\...
- 121