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A branch of geometry dealing with convex sets and functions. Polytopes, convex bodies, discrete geometry, linear programming, antimatroids, ...

2 votes
Accepted

Lipschitz constant of cental projection of unit ball to surface of convex body

$r=1$, $R=OC/OB$. Unit ball is $DEBD'$ and symmetric part, $M$ is $DCD'$ and symmetric part, $x=B$, $y=G$, $\pi(x)=C$, $\pi(y)=H$, ray $OGH$ is very close to $OBC$. We have $$\|C-H\|:\|B-G\|=\frac{C …
Fedor Petrov's user avatar
6 votes
Accepted

Internal edges in Convex Polytopes

If $n\geq 4$, let $S$ be a moment curve $f(t)=(t,t^2,\dots,t^n),t\in \mathbb{R}$. Any hyperplane contains at most $n$ points from $S$, since polynomial of degree at most $n$ has at most $n$ roots. So, …
Fedor Petrov's user avatar
2 votes
Accepted

Maximum number of points in convex position on a grid

Your guess is correct. There are at most two vertical sections which contain more than 2 points. So, totally we have at most $m+m+(n-2)2=2m+2n-4$ points. The example is the perimeter of the grid.
Fedor Petrov's user avatar
3 votes

Finite dimensional subspaces of $L^1.$

Greg is right, of course, the dual must be a zonotope. Let me mention also a direct characterization: for any vectors $x_1,\dots,x_k,y_1,\dots,y_m$ such that $\sum |f(x_i)|\geq \sum |f(y_j)|$ for any …
Fedor Petrov's user avatar
1 vote

Let $X\subseteq\mathbb{R}^n$ and let $F$ be a face of $\mathop{\rm conv} X$. Then $F=\mathop...

This follows from the Lemma. If $F$ is a face of $conv(X)$, $x\in F$ and a finite subset $A\subset X$ is inclusion-minimal subset for which $x\in conv(A)$, then $A\subset F$. Proof. Induction in $|A …
Fedor Petrov's user avatar
4 votes

Number of Inner Diagonals of Convex Hulls of $n+2$ Points in Convex Configuration in $E^n$

If $p_1,\dots,p_{n+2}$ are the vertices and $\sum \lambda_i p_i=0$, $\sum \lambda_i=0$, is their (let it be unique for a moment) affine dependence, the point $q=\alpha p_i+(1-\alpha) p_j$ on the diago …
Fedor Petrov's user avatar
14 votes
Accepted

covering convex sets by round balls

Yes. Any point $y$ in the convex hull of $x$'s is a barycenter of some non-negative masses $m_i$ in $x_i$, $\sum m_i=1$, $y=\sum m_i x_i$. Point $y$ minimizes the moment of inertia $I(p)=\sum m_i |p-x …
Fedor Petrov's user avatar
2 votes
Accepted

On centrally symmetric convex figures on the hyperbolic plane

"Only if" part (for a symmetric region, bisectors are concurrent) is rather clear. Now "if" part. Assume that the common point of all bisectors exists, denote it by $O$. Clearly every chord $AOB$ thro …
Fedor Petrov's user avatar
2 votes

Extreme points of convex hull of Minkowski sum

It consists of points $p=a_i+b_j$ for which there exists a linear functional $h$ such that $h$ attains its maximum on $A$ in unique point $a_i$ and on $B$ in unique point $b_j$.
Fedor Petrov's user avatar
1 vote
Accepted

partition of a convex set into squares

Well, let me prove that the answer is negative even for triangle. Rotating coordinate system we may suppose that vertical lines are not parallel to sides of triangle and to sides of all squares (as th …
Fedor Petrov's user avatar
6 votes
Accepted

Convex body with affine-equivalent cross-sections

For given integers $n>2$ and $k\in \{2,3,\dots,n-1\}$ the question of Banach asks whether any $n$-dimensional real Banach space with isometric $k$-dimensional sections is a Hilbert space. For $k=2$ th …
Fedor Petrov's user avatar
1 vote
Accepted

Convex-like properties of the polar parametrization of the boundary a convex body on the plane

If $f(x)=\min\{s>0:x/s\in B\}$ is Minkowski functional of $B$, then $f$ is a convex function on the plane and ${\bf p}(t)=\frac{e^{it}}{f(e^{it})}$. I think your claims now follow from the properties …
Fedor Petrov's user avatar
2 votes
Accepted

A claim on the concurrency of area bisectors of planar convex regions

A planar convex region is centrally symmetric if and only if its area bisectors are all concurrent. Yes. Let all area bisectors of our region $K$ pass through a point $O$. Note that any line $\ell$ …
Fedor Petrov's user avatar
3 votes

Monotonicity of perimeter of convex subsets of hyperbolic plane

Does not an elementary Euclidean proof work verbatim? If both sets are polygons, the interior one is obtained from the exterior one by cuts (i.e. transformations $P\to P\cap H$, where $P$ is a polygon …
Fedor Petrov's user avatar
3 votes
Accepted

Is a ball the hardest body to approximate by polytopes (in the Banach–Mazur metric)?

Let me give a proof of rozu's guess that $\phi_3(D)=1/2$ and $\phi_3(S)=\sqrt{2}-1$ if $D$, $S$ are a disk and a square. This disproves your conjecture. First of all, for every $K$ and every $N$ the c …
Fedor Petrov's user avatar

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