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We define and investigate a generalization of the notion of convex compacta. Namely, for semiconvex combination in a semiconvex compactum we allow the existence of non-trivial loops connecting a point with itself. It is proved that any semiconvex compactum contains two non-empty convex compacta, the center and the weak center. The center is the largest compactum such that semiconvex combination induces a convex structure on it. The convex structure on the weak center does not necessarily coincide...

Semi-monotone sets

Saugata Basu, Andrei Gabrielov, Nicolai Vorobjov (2013)

Journal of the European Mathematical Society

A coordinate cone in $ℝ^{n}$ is an intersection of some coordinate hyperplanes and open coordinate half-spaces. A semi-monotone set is an open bounded subset of $ℝ^{n}$ , definable in an o-minimal structure over the reals, such that its intersection with any translation of any coordinate cone is connected. This notion can be viewed as a generalization of convexity. Semi-monotone sets have a number of interesting geometric and combinatorial properties. The main result of the paper is that every semi-monotone...

Separating Convex Sets in the Plane.

J. Urrutia, J. Czyzowicz, E. Rivera-Campo, J. Zaks (1992)

Discrete & computational geometry

Separating plane convex sets.

Meir Katchalski, Rafael Hope (1990)

Mathematica Scandinavica

Separating Two Simple Polygons by a Sequence of Translations.

R. Pollack, M. Sharir, S. Sifrony (1988)

Discrete & computational geometry

Separation by hyperplanes in finite-dimensional vector spaces over Archimedean ordered fields.

Gritzmann, Peter, Klee, Victor (1998)

Journal of Convex Analysis

Séparation forte $k$ -branlante de deux convexes

Jacques Bair (1977)

Commentationes Mathematicae Universitatis Carolinae

Separation of $(n + 1)$ -families of sets in general position in $𝐑^{n}$

Mircea Balaj (1997)

Commentationes Mathematicae Universitatis Carolinae

In this paper the main result in [1], concerning $(n + 1)$ -families of sets in general position in $𝐑^{n}$ , is generalized. Finally we prove the following theorem: If ${A_{1}, A_{2}, \dots, A_{n + 1}}$ is a family of compact convexly connected sets in general position in $𝐑^{n}$ , then for each proper subset $I$ of ${1, 2, \dots, n + 1}$ the set of hyperplanes separating $\cup {A_{i} : i \in I}$ and $\cup {A_{j} : j \in \bar{I}}$ is homeomorphic to $S_{n}^{+}$ .

Separation of two convex sets by operators

Karl-Heinz Elster, Reinhard Nehse (1978)

Commentationes Mathematicae Universitatis Carolinae

Sets Expressible as Unions of Staircase $n$ -Convex Polygons

Marilyn Breen (2011)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Let $k$ and $n$ be fixed, $k \geq 1$ , $n \geq 1$ , and let $S$ be a simply connected orthogonal polygon in the plane. For $T \subseteq S, T$ lies in a staircase $n$ -convex orthogonal polygon $P$ in $S$ if and only if every two points of $T$ see each other via staircase $n$ -paths in $S$ . This leads to a characterization for those sets $S$ expressible as a union of $k$ staircase $n$ -convex polygons $P_{i}$ , $1 \leq i \leq k$ .

Sets invariant under projections onto one dimensional subspaces

Simon Fitzpatrick, Bruce Calvert (1991)

Commentationes Mathematicae Universitatis Carolinae

The Hahn–Banach theorem implies that if $m$ is a one dimensional subspace of a t.v.s. $E$ , and $B$ is a circled convex body in $E$ , there is a continuous linear projection $P$ onto $m$ with $P (B) \subseteq B$ . We determine the sets $B$ which have the property of being invariant under projections onto lines through $0$ subject to a weak boundedness type requirement.

Sets invariant under projections onto two dimensional subspaces

Simon Fitzpatrick, Bruce Calvert (1991)

Commentationes Mathematicae Universitatis Carolinae

The Blaschke–Kakutani result characterizes inner product spaces $E$ , among normed spaces of dimension at least 3, by the property that for every 2 dimensional subspace $F$ there is a norm 1 linear projection onto $F$ . In this paper, we determine which closed neighborhoods $B$ of zero in a real locally convex space $E$ of dimension at least 3 have the property that for every 2 dimensional subspace $F$ there is a continuous linear projection $P$ onto $F$ with $P (B) \subseteq B$ .

Sets of constant width and diametrically complete setes in normed spaces

Leoni Dalla, N. K. Tamvakis (1985)

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

Shaking compact sets.

Campi, Stefano, Colesanti, Andrea, Gronchi, Paolo (2001)

Beiträge zur Algebra und Geometrie

Sharpening on Mircea's inequality.

Wu, Yu-Dong, Zhang, Zhi-Hua, Lokesha, V. (2007)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

Shift inequalities of Gaussian type and norms of barycentres

F. Barthe, D. Cordero-Erausquin, M. Fradelizi (2001)

Studia Mathematica

We derive the equivalence of different forms of Gaussian type shift inequalities. This completes previous results by Bobkov. Our argument strongly relies on the Gaussian model for which we give a geometric approach in terms of norms of barycentres. Similar inequalities hold in the discrete setting; they improve the known results on the so-called isodiametral problem for the discrete cube. The study of norms of barycentres for subsets of convex bodies completes the exposition.

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