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1. CMB Online first
Plane Lorentzian and Fuchsian Hedgehogs Parts of the Brunn-Minkowski theory can be extended to hedgehogs, which are
envelopes of families of affine hyperplanes parametrized by their Gauss map.
F. Fillastre introduced Fuchsian convex bodies, which are the
closed convex sets of Lorentz-Minkowski space that are globally invariant
under the action of a Fuchsian group. In this paper, we undertake a study of
plane Lorentzian and Fuchsian hedgehogs. In particular, we prove the
Fuchsian analogues of classical geometrical inequalities (analogues which
are reversed as compared to classical ones).
Keywords:Fuchsian and Lorentzian hedgehogs, evolute, duality, convolution, reversed isoperimetric inequality, reversed Bonnesen inequality Categories:52A40, 52A55, 53A04, 53B30 |
2. CMB 2009 (vol 52 pp. 349)
On Projection Bodies of Order One The projection body of order one $\Pi_1K$ of a convex body $K$ in
$\R^n$ is the body whose support function is, up to a constant, the
average mean width of the orthogonal projections of $K$ onto
hyperplanes through the origin.
The paper contains an inequality for the support function of
$\Pi_1K$, which implies in particular that such a function is
strictly convex, unless $K$ has dimension one or two. Furthermore,
an existence problem related to the reconstruction of a convex body
is discussed to highlight the different behavior of the area
measures of order one and of order $n-1$.
Category:52A40 |
3. CMB 2009 (vol 52 pp. 380)
Successive Minima and Radii In this note we present inequalities relating the successive minima of an
$o$-symmetric convex body and the successive inner and outer radii of the
body. These inequalities join known inequalities involving only either
the successive minima or the successive radii.
Keywords:successive minima, inner and outer radii Categories:52A20, 52C07, 52A40, 52A39 |
4. CMB 2009 (vol 52 pp. 464)
Two Volume Product Inequalities and Their Applications Let $K \subset {\mathbb{R}}^{n+1}$ be a convex body of class $C^2$
with everywhere positive Gauss curvature. We show that there exists
a positive number $\delta (K)$ such that for any $\delta \in (0,
\delta(K))$ we have $\Volu(K_{\delta})\cdot
\Volu((K_{\delta})^{\sstar}) \geq \Volu(K)\cdot \Volu(K^{\sstar}) \geq
\Volu(K^{\delta})\cdot \Volu((K^{\delta})^{\sstar})$, where $K_{\delta}$,
$K^{\delta}$ and $K^{\sstar}$ stand for the convex floating body, the
illumination body, and the polar of $K$, respectively. We derive a
few consequences of these inequalities.
Keywords:affine invariants, convex floating bodies, illumination bodies Categories:52A40, 52A38, 52A20 |
5. CMB 2006 (vol 49 pp. 185)
On the Inequality for Volume and Minkowskian Thickness Given a centrally symmetric convex body $B$ in $\E^d,$ we denote
by $\M^d(B)$ the Minkowski space ({\em i.e.,} finite dimensional
Banach space) with unit ball $B.$ Let $K$ be an arbitrary convex
body in $\M^d(B).$ The relationship between volume $V(K)$ and the
Minkowskian thickness ($=$ minimal width) $\thns_B(K)$ of $K$ can
naturally be given by the sharp geometric inequality $V(K) \ge
\alpha(B) \cdot \thns_B(K)^d,$ where $\alpha(B)>0.$ As a simple
corollary of the Rogers--Shephard inequality we obtain that
$\binom{2d}{d}{}^{-1} \le \alpha(B)/V(B) \le 2^{-d}$ with equality
on the left attained if and only if $B$ is the difference body of
a simplex and on the right if $B$ is a cross-polytope. The main
result of this paper is that for $d=2$ the equality on the right
implies that $B$ is a parallelogram. The obtained results yield
the sharp upper bound for the modified Banach--Mazur distance to the
regular hexagon.
Keywords:convex body, geometric inequality, thickness, Minkowski space, Banach space, normed space, reduced body, Banach-Mazur compactum, (modified) Banach-Mazur distance, volume ratio Categories:52A40, 46B20 |
6. CMB 2003 (vol 46 pp. 373)
Potential Theory of the Farthest-Point Distance Function We study the farthest-point distance function, which measures the
distance from $z \in \mathbb{C}$ to the farthest point or points of
a given compact set $E$ in the plane.
The logarithm of this distance is subharmonic as a function of $z$,
and equals the logarithmic potential of a unique probability measure
with unbounded support. This measure $\sigma_E$ has many interesting
properties that reflect the topology and geometry of the compact set
$E$. We prove $\sigma_E(E) \leq \frac12$ for polygons inscribed in a
circle, with equality if and only if $E$ is a regular $n$-gon for some
odd $n$. Also we show $\sigma_E(E) = \frac12$ for smooth convex sets of
constant width. We conjecture $\sigma_E(E) \leq \frac12$ for all~$E$.
Keywords:distance function, farthest points, subharmonic function, representing measure, convex bodies of constant width Categories:31A05, 52A10, 52A40 |