Search Results

$r=1$, $R=OC/OB$. 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 …

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, …

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.

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 …

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 …

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 …

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 …

"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 …

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 $b_j$.

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 …

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 …

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 …

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 $\ell$ …

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 …

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 …