Veranstaltungen

04. Dezember 2025, 16:00 bis 18:00

AKOR Seminar: Polar Convexity in Finite Dimensional Euclidean Spaces

Seminar

Hristo Sendov from the Department of Statistical and Actuarial Sciences and the Department of Mathematics,

The University of Western Ontario, Canada

 

Let $\hat{\mathbb{R}}^n$ be the one point compactification of $\mathbb{R}^n$ obtained by adding a point at infinity.  We say that a subset $A\subseteq \hat{\mathbb{R}}^n$ is {\it $\mathbf{u}$-convex} if for every pair of points $\mathbf{z}_1, \mathbf{z}_2 \in A$, the arc of the unique circle through $\mathbf{u}, \mathbf{z}_1$ and $\mathbf{z}_2$, from $\mathbf{z}_1$ to $\mathbf{z}_2$ and not containing $\mathbf{u}$, is contained in $A$. In this case, we call $\mathbf{u}$ a pole of $A$. When the pole $\mathbf{u}$ approaches infinity, $\mathbf{u}$-convex sets become convex in the classical sense. 

 

The notion of polar convexity in the complex plane has been used to analyze the behavior of critical points of polynomials. 

In this talk, we extend the notion to finite dimensional Euclidean spaces. The goal of this work is to start building the theory of polar convexity and to show that the introduction of a pole creates a richer theory. For example, polar convexity enjoys a beautiful duality that does not exist in classical convexity. We formulate polar analogues of several classical results of the alternatives, such as Gordon and Farkas lemmas. Finally, we give a full description of the convex hull of finitely many points with respect to finitely many poles.

This is a joint work with Shubhankar Bhatt from the Department of Mathematics, The University of Western Ontario, Canada

Kalendereintrag

Veranstaltung Details

Veranstaltungsort
Sem. R. DB gelb 04
1040 Wien
Wiedner Hauptstraße 8 E105-4
Veranstalter
VADOR
vador@tuwien.ac.at
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