Stability Theorems for Convex Domains of Constant Width

1988 ◽  
Vol 31 (3) ◽  
pp. 328-337 ◽  
Author(s):  
H. Groemer
Keyword(s):  
Convex Domain ◽  
Hausdorff Distance ◽  
Stability Problem ◽  
Constant Width ◽  
Circular Disc ◽  
Convex Domains ◽  
Plane Convex ◽  
Reuleaux Triangle ◽  
Stability Theorems ◽  
Maximal Area

AbstractIt is known that among all plane convex domains of given constant width Reuleaux triangles have minimal and circular discs have maximal area. Some estimates are given concerning the following associated stability problem: If K is a convex domain of constant width w and if the area of K differs at most ∊ from the area of a Reuleaux triangle or a circular disc of width w, how close (in terms of the Hausdorff distance) is K to a Reuleaux triangle or a circular disc? Another result concerns the deviation of a convex domain M of diameter d from a convex domain of constant width if the perimeter of M is close to πd.

scholarly journals Symmetric homogeneous convex domains

1984 ◽  
Vol 93 ◽  
pp. 1-17
Author(s):  
Tadashi Tsuji
Keyword(s):  
Real Number ◽  
Convex Domain ◽  
Convex Cone ◽  
Affine Transformations ◽  
Convex Domains ◽  
Affine Line ◽  
Homogeneous Convex

Let D be a convex domain in the n-dimensional real number space Rn, not containing any affine line and A(D) the group of all affine transformations of Rn leaving D invariant. If the group A(D) acts transitively on D, then the domain D is said to be homogeneous. From a homogeneous convex domain D in Rn, a homogeneous convex cone V = V(D) in Rn+1 = Rn × R is constructed as follows (cf. Vinberg [11]):which is called the cone fitted on the convex domain D.

Maximal Areas of Reuleaux Polygons

1970 ◽  
Vol 13 (2) ◽  
pp. 175-179 ◽  
Author(s):  
G. T. Sallee
Keyword(s):  
Finite Number ◽  
Convex Set ◽  
Constant Width ◽  
Plane Convex ◽  
Circular Arcs ◽  
Positive Length ◽  
Reuleaux Polygon ◽  
Reuleaux Polygons

In this paper we provide new proofs of some interesting results of Firey [2] on isoperimetric ratios of Reuleaux polygons. Recall that a Reuleaux polygon is a plane convex set of constant width whose boundary consists of a finite (odd) number of circular arcs. Equivalently, it is the intersection of a finite number of suitably chosen congruent discs. For more details, see [1, p. 128].If a Reuleaux polygon has n sides (arcs) of positive length (where n is odd and ≥ 3), we will refer to it as a Reuleaux n-gon, or sometimes just as an n-gon. If all of the sides are equal, it is termed a regular n-gon.

SOLUTIONS TO ${\bar\partial}$-EQUATION ON STRONGLY q-CONVEX DOMAINS WITH Lp-ESTIMATES

2004 ◽  
Vol 01 (06) ◽  
pp. 739-749 ◽  
Author(s):  
OSAMA ABDELKADER ◽  
SHABAN KHIDR
Keyword(s):  
Vector Bundle ◽  
Convex Domain ◽  
Holomorphic Vector Bundle ◽  
Kähler Manifold ◽  
Convex Domains ◽  
Lp Estimates ◽  
Type K ◽  
Kahler Manifold

The purpose of this paper is to construct a solution with Lp-estimates, 1≤p≤∞, to the equation [Formula: see text] on strongly q-convex domain of Kähler manifold. This is done for forms of type (n,s), s≥ max (q,k), with values in a holomorphic vector bundle which is Nakano semi-positive of type k and for forms of type (0,s), q≤s≤n-k, with values in a holomorphic vector bundle which is Nakano semi-negative of type k.

Saturated systems of symmetric convex domains; results of Eggleston, Bambah and Woods

1973 ◽  
Vol 74 (1) ◽  
pp. 107-116 ◽  
Author(s):  
Vishwa Chander Dumir ◽  
Dharam Singh Khassa
Keyword(s):  
Lower Bound ◽  
Lebesgue Measure ◽  
Convex Domain ◽  
Gauge Function ◽  
Symmetric Convex ◽  
Convex Domains ◽  
Strictly Convex ◽  
Point Set ◽  
Image Position

Let K be a closed, bounded, symmetric convex domain with centre at the origin O and gauge function F(x). By a homothetic translate of K with centre a and radius r we mean the set {x: F(x−a) ≤ r}. A family ℳ of homothetic translates of K is called a saturated family or a saturated system if (i) the infimum r of the radii of sets in ℳ is positive and (ii) every homothetic translate of K of radius r intersects some member of ℳ. For a saturated family ℳ of homothetic translates of K, let S denote the point-set union of the interiors of members of ℳ and S(l), the set S ∪ {x: F(x) ≤ l}. The lower density ρℳ(K) of the saturated system ℳ is defined bywhere V(S(l)) denotes the Lebesgue measure of the set S(l). The problem is to find the greatest lower bound ρK of ρℳ(K) over all saturated systems ℳ of homothetic translates of K. In case K is a circle, Fejes Tóth(9) conjectured thatwhere ϑ(K) denotes the density of the thinnest coverings of the plane by translates of K. In part I, we state results already known in this direction. In part II, we prove that ρK = (¼) ϑ(K) when K is strictly convex and in part III, we prove that ρK = (¼) ϑ(K) for all symmetric convex domains.

scholarly journals Zero varieties for the Nevanlinna class on all convex domains of finite type

2001 ◽  
Vol 163 ◽  
pp. 215-227 ◽  
Author(s):  
Klas Diederich ◽  
Emmanuel Mazzilli
Keyword(s):  
Finite Type ◽  
Convex Domain ◽  
Main Tool ◽  
Smooth Convex ◽  
Convex Domains ◽  
Zero Sets ◽  
Nevanlinna Class ◽  
Cauchy Riemann ◽  
Bounded Smooth

It is shown, that the so-called Blaschke condition characterizes in any bounded smooth convex domain of finite type exactly the divisors which are zero sets of functions of the Nevanlinna class on the domain. The main tool is a non-isotropic L1 estimate for solutions of the Cauchy-Riemann equations on such domains, which are obtained by estimating suitable kernels of Berndtsson-Andersson type.

Stability for the Minkowski measure of convex domains of constant width

2013 ◽  
Vol 104 (3) ◽  
pp. 505-513
Author(s):  
HaiLin Jin ◽  
GangSong Leng ◽  
Qi Guo
Keyword(s):  
Constant Width ◽  
Convex Domains

scholarly journals APPROXIMATION OF HOLOMORPHIC MAPPINGS ON 1-CONVEX DOMAINS

2013 ◽  
Vol 24 (14) ◽  
pp. 1350108 ◽  
Author(s):  
KRIS STOPAR
Keyword(s):  
Complex Manifold ◽  
Convex Domain ◽  
Holomorphic Mappings ◽  
Holomorphic Maps ◽  
Convex Domains ◽  
Exceptional Set ◽  
Proper Holomorphic Maps ◽  
Pseudoconvex Boundary ◽  
Oka Principle ◽  
Additional Assumptions

Let π : Z → X be a holomorphic submersion of a complex manifold Z onto a complex manifold X and D ⋐ X a 1-convex domain with strongly pseudoconvex boundary. We prove that under certain conditions there always exists a spray of π-sections over [Formula: see text] which has prescribed core, it fixes the exceptional set E of D, and is dominating on [Formula: see text]. Each section in this spray is of class [Formula: see text] and holomorphic on D. As a consequence we obtain several approximation results for π-sections. In particular, we prove that π-sections which are of class [Formula: see text] and holomorphic on D can be approximated in the [Formula: see text] topology by π-sections that are holomorphic in open neighborhoods of [Formula: see text]. Under additional assumptions on the submersion we also get approximation by global holomorphic π-sections and the Oka principle over 1-convex manifolds. We include an application to the construction of proper holomorphic maps of 1-convex domains into q-convex manifolds.

DILATATION AND ORDER OF CONTACT FOR HOLOMORPHIC SELF-MAPS OF STRONGLY CONVEX DOMAINS

2003 ◽  
Vol 86 (1) ◽  
pp. 131-152 ◽  
Author(s):  
FILIPPO BRACCI
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Convex Domain ◽  
Lower Semicontinuous ◽  
Subject Classification ◽  
Convex Domains ◽  
Strongly Convex ◽  
Intrinsic Measure

Let $D$ be a bounded strongly convex domain and let $f$ be a holomorphic self-map of $D$. In this paper we introduce and study the dilatation $\alpha (f)$ of $f$ defined, if $f$ has no fixed points in $D$, as the usual boundary dilatation coefficient of $f$ at its Wolff point, or, if $f$ has some fixed points in $D$, as the ratio of shrinking of the Kobayashi balls around a fixed point of $f$. In particular, we show that the map $\alpha$, defined as $\alpha : f \mapsto \alpha (f) \in [0,1]$, is lower semicontinuous. Among other things, this allows us to study the limits of a family of holomorphic self-maps of $D$. In the case of an inner fixed point, the dilatation is an intrinsic measure of the order of contact of $f(D)$ to $\partial D$.Finally, using complex geodesics, we define and study a directional dilatation, which is a measure of the shrinking of the domain along a given direction. Again, results of semicontinuity are given and applied to a family of holomorphic self-maps.2000 Mathematical Subject Classification: primary 32H99; secondary 30F99, 32H15.

SOLUTION TO $\overline\partial$ AND $\overline\partial_{b}$ PROBLEM FOR SMOOTH FORMS AND CURRENTS ON STRICTLY q-CONVEX DOMAINS

2012 ◽  
Vol 09 (04) ◽  
pp. 1220002
Author(s):  
SAYED SABER
Keyword(s):  
Convex Domain ◽  
Kähler Manifold ◽  
Convex Domains ◽  
Kahler Manifold

On a strictly q-convex domain D in a Kähler manifold X, we obtain the solvability of the [Formula: see text]-problem for smooth forms and currents on boundaries of D. Moreover, we study the solvability of the [Formula: see text]-problem for extensible currents.

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