Re: Winternitz Theorem
From: Gerry Myerson (gerry_at_maths.mq.edi.ai.i2u4email)
Date: 09/24/04
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Date: Fri, 24 Sep 2004 14:00:06 +0000 (UTC)
In article <civrim$kmg$1@news.ks.uiuc.edu>,
andreas wagener <108076@gmx.net> wrote:
> The following nice theorem (phrased as an exercise) appears in
> Yaglom/Boltyanskii's book "Convex Figures" (p. 36):
>
> "A convex figure is divided into two parts by a line L that passes
> through its center of gravity. Prove that the ratio of the areas of
> the two parts always lies between the bounds 4/5 and 5/4."
>
> Yaglom and Boltyanskii called this "Winternitz' Theorem". However,
> they do not provide an original reference, and I was not able to trace
> back the original source for this theorem either.
>
> Can anybody help me? Is there more literature on Winternitz' Theorem?
Here's something from Math Reviews that may be of use.
MR1099773 (92e:52010)
Scott, P. R.(5-ADLD)
On the union of convex bodies with no interior point in common.
Mathematika 37 (1990), no. 2, 245--250.
52A35 (52A20)
Let $K_1,\cdots,K_{d+1}$ be $d+1$ convex bodies in $d$-dimensional
Euclidean space which have no interior point in common. The author shows
that $m(\bigcup_iK_i)\geq C_d\min_jm(K_j)$, where $i$ and $j$ vary from
$1$ to $d+1$, $C_d$ is a constant depending only on the dimension and
$m(·)$ is the measure of the set. This generalizes the author's earlier
result in the plane \ref[same journal 25 (1978), no. 1, 17--23;
MR0500525 (58 \#18140)]. Equality holds if and only if each $K_i$ is the
nonsimplicial set cut from a fixed $d$-simplex by one of the hyperplanes
through the center of gravity of the simplex and parallel to a facet of
the simplex. The proof relies on Radon's theorem and the $d$-dimensional
Winternitz theorem.
Reviewed by Jane R. Sangwine-Yager
-- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
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