Expand all Collapse all | Results 1 - 6 of 6 |
1. CJM 2010 (vol 62 pp. 1404)
Characterizations of Extremals for some Functionals on Convex Bodies
We investigate equality cases in inequalities for Sylvester-type
functionals. Namely, it was proven by Campi, Colesanti, and Gronchi
that the quantity
$$
\int_{x_0\in K}\cdots\int_{x_n\in
K}[V(\textrm{conv}\{x_0,\dots,x_n\})]^pdx_0\cdots dx_n , n\geq d, p\geq
1
$$
is maximized by triangles among all planar convex bodies $K$
(parallelograms in the symmetric case). We show that these are the
only maximizers, a fact proven by Giannopoulos for $p=1$.
Moreover, if $h$: $\mathbb{R}_+\rightarrow \mathbb{R}_+$ is a
strictly increasing function and $W_j$ is the $j$-th
quermassintegral in $\mathbb{R}^d$, we prove that the functional
$$
\int_{x_0\in K_0}\cdots\int_{x_n\in
K_n}h(W_j(\textrm{conv}\{x_0,\dots,x_n\}))dx_0\cdots dx_n , n \geq d
$$
is
minimized among the $(n+1)$-tuples of convex bodies of fixed
volumes if and only if $K_0,\dots,K_n$ are homothetic ellipsoids
when $j=0$ (extending a result of Groemer) and Euclidean balls
with the same center when $j>0$ (extending a result of Hartzoulaki
and Paouris).
Categories:52A40, 52A22 |
2. CJM 2008 (vol 60 pp. 3)
Convex Bodies of Minimal Volume, Surface Area and Mean Width with Respect to Thin Shells Given $r>1$, we consider convex bodies in $\E^n$ which
contain a fixed unit ball, and whose
extreme points are of distance at least $r$ from the centre of
the unit ball, and we investigate how well these
convex bodies approximate the unit ball in terms of volume, surface area and
mean width. As $r$ tends to one, we prove asymptotic formulae
for the error of the approximation, and provide good estimates on
the involved constants depending on the dimension.
Categories:52A27, 52A40 |
3. CJM 2006 (vol 58 pp. 600)
Geometric Study of Minkowski Differences of Plane Convex Bodies In the Euclidean plane $\mathbb{R}^{2}$, we define the Minkowski difference
$\mathcal{K}-\mathcal{L}$ of two arbitrary convex bodies $\mathcal{K}$,
$\mathcal{L}$ as a rectifiable closed curve $\mathcal{H}_{h}\subset \mathbb{R}
^{2}$ that is determined by the difference $h=h_{\mathcal{K}}-h_{\mathcal{L}
} $ of their support functions. This curve $\mathcal{H}_{h}$ is
called the
hedgehog with support function $h$. More generally, the object of hedgehog
theory is to study the Brunn--Minkowski theory in the vector space of
Minkowski differences of arbitrary convex bodies of Euclidean space $\mathbb{R}
^{n+1}$, defined as (possibly singular and self-intersecting) hypersurfaces
of $\mathbb{R}^{n+1}$. Hedgehog theory is useful for: (i)
studying convex bodies by splitting them into a sum in order to reveal their
structure; (ii) converting analytical problems into
geometrical ones by considering certain real functions as support
functions.
The purpose of this paper is to give a detailed study of plane
hedgehogs, which constitute the basis of the theory. In particular:
(i) we study their length measures and solve the extension of the
Christoffel--Minkowski problem to plane hedgehogs; (ii) we
characterize support functions of plane convex bodies among support
functions of plane hedgehogs and support functions of plane hedgehogs among
continuous functions; (iii) we study the mixed area of
hedgehogs in $\mathbb{R}^{2}$ and give an extension of the classical Minkowski
inequality (and thus of the isoperimetric inequality) to hedgehogs.
Categories:52A30, 52A10, 53A04, 52A38, 52A39, 52A40 |
4. CJM 2004 (vol 56 pp. 529)
Asymptotics for Minimal Discrete Riesz Energy on Curves in $\R^d$ We consider the $s$-energy
$$
E(\ZZ_n;s)=\sum_{i \neq j} K(\|z_{i,n}-z_{j,n}\|;s)
$$
for point sets $\ZZ_n=\{ z_{k,n}:k=0,\dots,n\}$ on certain compact sets
$\Ga$ in $\R^d$ having finite one-dimensional Hausdorff measure, where
$$
K(t;s)=
\begin{cases}
t^{-s} ,& \mbox{if } s>0, \\
-\ln t, & \mbox{if } s=0,
\end{cases}
$$
is the Riesz kernel. Asymptotics for the minimum $s$-energy and the
distribution of minimizing sequences of points is studied. In
particular, we prove that, for $s\geq 1$, the minimizing nodes for a
rectifiable Jordan curve $\Ga$ distribute asymptotically uniformly with
respect to arclength as $n\to\infty$.
Keywords:Riesz energy, Minimal discrete energy,, Rectifiable curves, Best-packing on curves Categories:52A40, 31C20 |
5. CJM 1999 (vol 51 pp. 449)
A Brunn-Minkowski Type Theorem on the Minkowski Spacetime In this article, we derive a Brunn-Minkowski type theorem
for sets bearing some relation to the causal structure
on the Minkowski spacetime $\mathbb{L}^{n+1}$. We also
present an isoperimetric inequality in the Minkowski
spacetime $\mathbb{L}^{n+1}$ as a consequence of this
Brunn-Minkowski type theorem.
Keywords:Minkowski spacetime, Brunn-Minkowski inequality, isoperimetric inequality Categories:53B30, 52A40, 52A38 |
6. CJM 1997 (vol 49 pp. 1162)
Isoperimetric inequalities on surfaces of constant curvature In this paper we introduce the concepts of hyperbolic and elliptic
areas and prove uncountably many new geometric isoperimetric
inequalities on the surfaces of constant curvature.
Keywords:Gaussian curvature, Gauss-Bonnet theorem, polygon, pseudo-polygon, pseudo-perimeter, hyperbolic surface, Heron's formula, analytic and geometric isoperimetric inequalities Categories:51M10, 51M25, 52A40, 53C20 |