Locally Convex Hypersurfaces

1973 ◽  
Vol 25 (3) ◽  
pp. 531-538 ◽  
Author(s):  
L. B. Jonker ◽  
R. D. Norman
Keyword(s):  
Convex Body ◽  
Open Subset ◽  
Convex Hypersurface ◽  
Topological Manifold ◽  
Continuous Map ◽  
Open Set ◽  
Locally Convex ◽  
Convex Hypersurfaces

Let M be an n-dimensional connected topological manifold. Let ξ : M → Rn+1 be a continuous map with the following property: to each x ∈ M there is an open set x ∈ Ux ⊂ M, and a convex body Kx ⊂ Rn+1 such that ξ(UX) is an open subset of ∂Kx and such that is a homeomorphism onto its image. We shall call such a mapping ξ a locally convex immersion and, along with Van Heijenoort [8] we shall call ξ(M) a locally convex hypersurface of Rn+1.

Prescribing Centro-Affine Curvature From One Convex Body to Another

2020 ◽  
Author(s):  
Alina Stancu
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
Minkowski Inequality ◽  
Curvature Flow ◽  
Constant Volume ◽  
Convex Hypersurface ◽  
Strictly Convex ◽  
Initial Hypersurface ◽  
Affine Curvature ◽  
Convex Hypersurfaces

Abstract We study a curvature flow on smooth, closed, strictly convex hypersurfaces in $\mathbb{R}^n$, which commutes with the action of $SL(n)$. The flow shrinks the initial hypersurface to a point that, if rescaled to enclose a domain of constant volume, is a smooth, closed, strictly convex hypersurface in $\mathbb{R}^n$ with centro-affine curvature proportional, but not always equal, to the centro-affine curvature of a fixed hypersurface. We outline some consequences of this result for the geometry of convex bodies and the logarithmic Minkowski inequality.

ON THE SPECTRALIFICATION OF A HEMISPECTRAL SPACE

2011 ◽  
Vol 10 (04) ◽  
pp. 687-699
Author(s):  
OTHMAN ECHI ◽  
MOHAMED OUELD ABDALLAHI
Keyword(s):  
Topological Space ◽  
Open Subset ◽  
Full Subcategory ◽  
Topological Spaces ◽  
Open Problems ◽  
Complete Answer ◽  
Reflective Subcategory ◽  
Continuous Map ◽  
Open Set ◽  
Spectral Spaces

An open subset U of a topological space X is called intersection compact open, or ICO, if for every compact open set Q of X, U ∩ Q is compact. A continuous map f of topological spaces will be called spectral if f-1 carries ICO sets to ICO sets. Call a topological space Xhemispectral, if the intersection of two ICO sets of X is an ICO. Let HSPEC be the category whose objects are hemispectral spaces and arrows spectral maps. Let SPEC be the full subcategory of HSPEC whose objects are spectral spaces. The main result of this paper proves that SPEC is a reflective subcategory of HSPEC. This gives a complete answer to Problem BST1 of "O. Echi, H. Marzougui and E. Salhi, Problems from the Bizerte–Sfax–Tunis seminar, in Open Problems in Topology II, ed. E. Pearl (Elsevier, 2007), pp. 669–674."

Non-hyperbolic P-Invariant Closed Characteristics on Partially Symmetric Compact Convex Hypersurfaces

2018 ◽  
Vol 18 (4) ◽  
pp. 763-774
Author(s):  
Hui Liu ◽  
Gaosheng Zhu
Keyword(s):  
Compact Convex ◽  
Convex Hypersurface ◽  
Closed Characteristics ◽  
Compact Convex Hypersurfaces ◽  
Partially Symmetric ◽  
Convex Hypersurfaces

AbstractLet {n\geq 2} be an integer, {P=\mathrm{diag}(-I_{n-\kappa},I_{\kappa},-I_{n-\kappa},I_{\kappa})} for some integer {\kappa\in[0,n]}, and let {\Sigma\subset{\mathbb{R}}^{2n}} be a partially symmetric compact convex hypersurface, i.e., {x\in\Sigma} implies {Px\in\Sigma}, and {(r,R)}-pinched. In this paper, we prove that when {{R/r}<\sqrt{5/3}} and {0\leq\kappa\leq[\frac{n-1}{2}]}, there exist at least {E(\frac{n-2\kappa-1}{2})+E(\frac{n-2\kappa-1}{3})} non-hyperbolic P-invariant closed characteristics on Σ. In addition, when {{R/r}<\sqrt{3/2}}, {[\frac{n+1}{2}]\leq\kappa\leq n} and Σ carries exactly nP-invariant closed characteristics, then there exist at least {2E(\frac{2\kappa-n-1}{4})+E(\frac{n-\kappa-1}{3})} non-hyperbolic P-invariant closed characteristics on Σ, where the function {E(a)} is defined as {E(a)=\min{\{k\in{\mathbb{Z}}\mid k\geq a\}}} for any {a\in\mathbb{R}}.

scholarly journals Extensions of ordered theories by generic predicates

2013 ◽  
Vol 78 (2) ◽  
pp. 369-387 ◽  
Author(s):  
Alfred Dolich ◽  
Chris Miller ◽  
Charles Steinhorn
Keyword(s):  
Open Subset ◽  
Connected Components ◽  
Linear Orders ◽  
Algebraic Numbers ◽  
Borel Set ◽  
Cartesian Power ◽  
Open Set ◽  
Unary Relation

Given a theory T extending that of dense linear orders without endpoints (DLO), in a language ℒ ⊇ {<}, we are interested in extensions T′ of T in languages extending ℒ by unary relation symbols that are each interpreted in models of T′ as sets that are both dense and codense in the underlying sets of the models.There is a canonically “wild” example, namely T = Th(〈ℝ, <, +, ·〉) and T′ = Th(〈ℝ, <, +, · ℚ 〉). Recall that T is o-minimal, and so every open set definable in any model of T has only finitely many definably connected components. But it is well known that 〈ℝ, <, +, · ℚ 〉 defines every real Borel set, in particular, every open subset of any finite cartesian power of ℝ and every subset of any finite cartesian power of ℚ. To put this another way, the definable open sets in models of T are essentially as simple as possible, while T′ has a model where the definable open sets are as complicated as possible, as is the structure induced on the new predicate.In contrast to the preceding example, if ℝalg is the set of real algebraic numbers and T′ Th(〈ℝ, <, +, ·, 〈alg〉), then no model of T′ defines any open set (of any arity) that is not definable in the underlying model of T.

scholarly journals THE CLOSED RANGE PROPERTY FOR BANACH SPACE OPERATORS

2008 ◽  
Vol 50 (1) ◽  
pp. 17-26 ◽  
Author(s):  
THOMAS L. MILLER ◽  
VLADIMIR MÜLLER
Keyword(s):  
Banach Space ◽  
Complex Plane ◽  
Open Subset ◽  
Weak Topology ◽  
Bounded Operator ◽  
Complex Banach Space ◽  
Fréchet Space ◽  
Closed Range ◽  
Open Set ◽  
Banach Space Operators

AbstractLetTbe a bounded operator on a complex Banach spaceX. LetVbe an open subset of the complex plane. We give a condition sufficient for the mappingf(z)↦ (T−z)f(z) to have closed range in the Fréchet spaceH(V,X) of analyticX-valued functions onV. Moreover, we show that there is a largest open setUfor which the mapf(z)↦ (T−z)f(z) has closed range inH(V,X) for allV⊆U. Finally, we establish analogous results in the setting of the weak–* topology onH(V, X*).

On α*-I-Open Set and α*-I-Continuous Map

2018 ◽  
Vol 56 (2) ◽  
pp. 101-103
Author(s):  
J. K. Ma itra ◽  
◽  
Rajesh Kumar Tiwari
Keyword(s):  
Continuous Map ◽  
Open Set

scholarly journals Curves with zero derivative in F-spaces

1981 ◽  
Vol 22 (1) ◽  
pp. 19-29 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Characteristic Function ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Continuous Map ◽  
Left And Right ◽  
The Mean ◽  
Image Position ◽  
Locally Convex

Let X be an F-space (complete metric linear space) and suppose g:[0, 1] → X is a continuous map. Suppose that g has zero derivative on [0, 1], i.e.for 0≤t≤1 (we take the left and right derivatives at the end points). Then, if X is locally convex or even if it merely possesses a separating family of continuous linear functionals, we can conclude that g is constant by using the Mean Value Theorem. If however X* = {0} then it may happen that g is not constant; for example, let X = Lp(0, 1) (0≤p≤1) and g(t) = l[0,t] (0≤t≤1) (the characteristic function of [0, t]). This example is due to Rolewicz [6], [7; p. 116].

Locally convex hypersurfaces immersed in $$\mathbb {H}^n\times \mathbb {R}$$ H n × R

2016 ◽  
Vol 188 (1) ◽  
pp. 17-32 ◽  
Author(s):  
Inês S. de Oliveira Padilha ◽  
S. J. Paul A. Schweitzer
Keyword(s):  
Locally Convex ◽  
Convex Hypersurfaces
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