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Nikolai Nikolov

Publications and source records attributed to Nikolai Nikolov.

At least 73 records · Page 4Linked to original sources
arXiv2012

On extremums of sums of powered distances to a finite set of points

In this paper we investigate the extremal properties of the sum $$\sum_{i=1}^n|MA_i|^λ,$$ where $A_i$ are vertices of a regular simplex, a cross-polytope (orthoplex) or a cube and $M$ varies on a sphere concentric to the sphere circumscribed around one of the given polytopes. We give full characterization for which points on $Γ$ the extremal values of the sum are obtained in terms of $λ$. In the case of the regular dodecahedron and icosahedron in $\mathbb{R}^3$ we obtain results for which values of $λ$ the corresponding sum is independent of the position of $M$ on $Γ$. We use elementary analytic and purely geometric methods.

math.MG↗
arXiv2012

Real and complex k-planes in convex hypersurfaces

It is shown that that the rank of the second fundamental form (resp. the Levi form) of a $\mathcal C^2$-smooth convex hypersurface $M$ in $\Bbb R^{n+1}$ (resp. $\Bbb C^{n+1}$) does not exceed an integer constant $k<n$ near a point $p\in M,$ then through any point $q\in M$ near $p$ there exists a real (resp. complex) $(n-k)$-dimensional plane that locally lies on $M.$

math.CV↗
arXiv2012

Convex domains with locally Levi-flat boundaries

It is shown that a domain in $\C^n$ which is locally convex and has $\mathcal C^1$-smooth Levi-flat boundary is locally linearly equivalent to a Cartesian product of a planar domain and $\C^{n-1}.$ This result does not extend to the case where the Levi form has locally constant rank for some $k<n-1$.

math.CV↗
arXiv2011

On the sum of powered distances to certain sets of points on the circle

In this paper we consider an extremal problem in geometry. Let $λ$ be a real number and $A$, $B$ and $C$ be arbitrary points on the unit circle $Γ$. We give full characterization of the extremal behavior of the function $f(M,λ)=MA^λ+MB^λ+MC^λ$, where $M$ is a point on the unit circle as well. We also investigate the extremal behavior of $\sum_{i=1}^nXP_i$, where $P_i, i=1,...,n$ are the vertices of a regular $n$-gon and $X$ is a point on $Γ$, concentric to the circle circumscribed around $P_1...P_n$. We use elementary analytic and purely geometric methods in the proof.

math.MG↗
arXiv2011

Two-dimensional slices of non-pseudoconvex open sets

Let $D$ be a non-pseudoconvex open set in $\C^3$ and $S$ be the union of all two-dimensional planes with non-empty and non-pseudoconvex intersection with $D.$ Sufficient conditions are given for $\C^3\setminus S$ to belong to a complex line. Moreover, in the $\mathcal C^2$-smooth case, it is shown that $S=\C^n$.

math.CV↗
arXiv2011

Rigid characterizations of pseudoconvex domains

We prove that an open set $D$ in $\C^n$ is pseudoconvex if and only if for any $z\in D$ the largest balanced domain centered at $z$ and contained in $D$ is pseudoconvex, and consider analogues of that characterization in the linearly convex case.

math.CV↗
arXiv2010

Estimates for invariant metrics near a non-semipositive boundary point

We find the precise growth of some invariant metrics near a point on the boundary of a domain where the Levi form has at least one negative eigenvalue. We also introduce a new invariant pseudometric which is convenient in this context, and give some of its general properties.

math.CV↗
arXiv2009

One-radius results for supermedian functions on $\Bbb R^d$, $d\le 2$

A classical result states that every lower bounded superharmonic function on $\Bbb R^2$ is constant. In this paper the following (stronger) one-circle version is proven. If $f\colon \Bbb R^2\to (-\infty,\infty]$ is lower semicontinuous, $\liminf_{|x|\to\infty} f(x)/\ln|x|\ge 0$, and, for every $x\in\Bbb R^2$, $1/(2π) \int_0^{2π} f(x+r(x)e^{it}) dt\le f(x)$, where $r\colon \Bbb R^2\to (0,\infty)$ is continuous, $\sup_{x\in\Bbb R^2} (r(x)-|x|)<\infty$, and $\inf_{x\in\Bbb R^2} (r(x)-|x|)=-\infty$, then $f$ is constant. Moreover, it is shown that, with respect to the assumption $r\le c|\cdot|+M$ on $\Bbb R^d$, there is a striking difference between the restricted volume mean property for the cases $d=1$ and $d=2$.

math.CV↗