Saddle point theorems on generalized convex spaces.
Schur-convexity for a class of symmetric functions and its applications.
Section and Projection Means of Convex Bodies.
Self-similar and Markov composition structures.
Semiconvex compacta
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
A coordinate cone in is an intersection of some coordinate hyperplanes and open coordinate half-spaces. A semi-monotone set is an open bounded subset of , 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.
Separating plane convex sets.
Separating Two Simple Polygons by a Sequence of Translations.
Separation by hyperplanes in finite-dimensional vector spaces over Archimedean ordered fields.
Séparation forte -branlante de deux convexes
Separation of -families of sets in general position in
In this paper the main result in [1], concerning -families of sets in general position in , is generalized. Finally we prove the following theorem: If is a family of compact convexly connected sets in general position in , then for each proper subset of the set of hyperplanes separating and is homeomorphic to .
Separation of two convex sets by operators
Sets Expressible as Unions of Staircase -Convex Polygons
Let and be fixed, , , and let be a simply connected orthogonal polygon in the plane. For lies in a staircase -convex orthogonal polygon in if and only if every two points of see each other via staircase -paths in . This leads to a characterization for those sets expressible as a union of staircase -convex polygons , .
Sets invariant under projections onto one dimensional subspaces
The Hahn–Banach theorem implies that if is a one dimensional subspace of a t.v.s. , and is a circled convex body in , there is a continuous linear projection onto with . We determine the sets which have the property of being invariant under projections onto lines through subject to a weak boundedness type requirement.
Sets invariant under projections onto two dimensional subspaces
The Blaschke–Kakutani result characterizes inner product spaces , among normed spaces of dimension at least 3, by the property that for every 2 dimensional subspace there is a norm 1 linear projection onto . In this paper, we determine which closed neighborhoods of zero in a real locally convex space of dimension at least 3 have the property that for every 2 dimensional subspace there is a continuous linear projection onto with .
Sets of constant width and diametrically complete setes in normed spaces
Shaking compact sets.
Sharpening on Mircea's inequality.
Shift inequalities of Gaussian type and norms of barycentres
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.