# Zelinskii Yu. B.

### Generalizations of the shadow problem

Stefanchuk M. V., Zelinskii Yu. B.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 6. - pp. 757-763

We solve the shadow problem in the n-dimensional Euclidean space $N^n$ for a family of sets obtained from any convex domain with nonempty interior with the help of parallel translations and homotheties. We determine the number of balls with centers on the sphere, sufficient for giving a shadow in the $n$-dimensional complex (hypercomplex) space.

### Generalized convex sets and the problem of shadow

Stefanchuk M. V., Vyhovs'ka I. Yu., Zelinskii Yu. B.

Ukr. Mat. Zh. - 2015. - 67, № 12. - pp. 1658-1666

The problem of shadow is solved. It is equivalent to the problem of finding conditions for a point to belong to a generalized convex hull of a family of compact sets.

### Development of Complex Analysis and Potential Theory at the Institute of Mathematics of the Ukrainian National Academy of Sciences in 1991–2013

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 763–779

### Theorems on Inclusion for Multivalued Mappings

Klishchuk B. A., Tkachuk M. V., Zelinskii Yu. B.

Ukr. Mat. Zh. - 2014. - 66, № 7. - pp. 1003–1005

The paper is devoted to the investigation of some properties of multivalued mappings in Euclidean spaces. Fixed-point theorems are proved for multivalued mappings whose restrictions to a certain subset in the closure of a domain satisfy a “coacute angle condition” or a “strict coacute angle condition.” Similar results for the restrictions of multivalued mappings satisfying certain metric conditions are also obtained.

### Yurii Ivanovych Samoilenko (on the 80th anniversary of his birthday)

Bakhtin A. K., Gerasimenko V. I., Plaksa S. A., Samoilenko A. M., Sharko V. V., Trohimchuk Yu. Yu, Yacenko V. O., Zelinskii Yu. B.

Ukr. Mat. Zh. - 2012. - 64, № 4. - pp. 574-576

### On some criteria of convexity for compact sets

Tkachuk M. V., Vyhovs'ka I. Yu., Zelinskii Yu. B.

Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 466-471

We establish some criteria of convexity of compact sets in the Euclidean space. Analogs of these results are extended to complex and hypercomplex cases.

### On a mapping of a projective space into a sphere

Ukr. Mat. Zh. - 2010. - 62, № 7. - pp. 937–944

We obtain an exact estimate for the minimum multiplicity of a continuous finite-to-one mapping of a projective space into a sphere for all dimensions. For finite-to-one mappings of a projective space into a Euclidean space, we obtain an exact estimate for this multiplicity for $n = 2, 3$. For $n ≥ 4$, we prove that this estimate does not exceed 4. Several open questions are formulated.

### Yuri Yurievich Trokhimchuk (on his 80th birthday)

Berezansky Yu. M., Bojarski B., Gorbachuk M. L., Kopilov A. P., Korolyuk V. S., Lukovsky I. O., Mitropolskiy Yu. A., Portenko N. I., Reshetnyak Yu. G., Samoilenko A. M., Sharko V. V., Shevchuk I. A., Skorokhod A. V., Tamrazov P. M., Zelinskii Yu. B.

Ukr. Mat. Zh. - 2008. - 60, № 5. - pp. 701 – 703

### Criterion for convexity of a domain of a Euclidean space

Vyhovs'ka I. Yu., Zelinskii Yu. B.

Ukr. Mat. Zh. - 2008. - 60, № 5. - pp. 709–711

We establish an external criterion for the convexity of a domain of a Euclidean space.

### On domains with regular sections

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1420–1423

We prove the generalized convexity of domains satisfying the condition of acyclicity of their sections by a certain continuously parametrized family of two-dimensional planes.

### Multiplicity of Continuous Mappings of Domains

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 554–558

We prove that either the proper mapping of a domain of an *n*-dimensional manifold onto a domain of another *n*-dimensional manifold of degree *k* is an interior mapping or there exists a point in the image that has at least |*k*|+2 preimages. If the restriction of *f* to the interior of the domain is a zero-dimensional mapping, then, in the second case, the set of points of the image that have at least |*k*|+2 preimages contains a subset of total dimension *n*. In addition, we construct an example of a mapping of a two-dimensional domain that is homeomorphic at the boundary and zero-dimensional, has infinite multiplicity, and is such that its restriction to a sufficiently large part of the branch set is a homeomorphism.

### On Locally Linearly Convex Domains

Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 280-284

We construct a counterexample to the hypothesis on global linear convexity of locally linearly convex domains with everywhere smooth boundary. We refine the theorem on the topological classification of linearly convex domains with smooth boundary.

### Helly Theorem and Related Results

Ukr. Mat. Zh. - 2002. - 54, № 1. - pp. 125-128

By using the classical Helly theorem, one cannot obtain information about a family of convex compact sets in the *n*-dimensional Euclidean space if it is known that only subfamilies consisting of *k* elements, 0 < *k* ≤ *n*, have nonempty intersections. We modify the Helly theorem to fix this issue and investigate the behavior of generalized convex families.

### On $(n, m)$-Convex Sets

Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 422-427

We investigate the class of generalized convex sets on Grassmann manifolds, which includes known generalizations of convex sets for Euclidean spaces. We extend duality theorems (of polarity type) to a broad class of subsets of the Euclidean space. We establish that the invariance of a mapping on generalized convex sets is equivalent to its affinity.

### Linearly convex regions with smooth boundaries

Ukr. Mat. Zh. - 1988. - 40, № 1. - pp. 53–58

### Derivative sets of Lipschitz functions having derivatives along certain directions

Ukr. Mat. Zh. - 1982. - 34, № 4. - pp. 421—427

### Application of local degree to the study of quasilight open mappings

Ukr. Mat. Zh. - 1978. - 30, № 3. - pp. 299–308

### An extension theorem and domain preservation criteria for multivalued maps

Ukr. Mat. Zh. - 1977. - 29, № 3. - pp. 383–387