Open Problem Garden

Gersgorin theorem states that: all the eigenvalues of $A=[a_{ij}]\in M_n$ are located in the union of $n$ discs $\bigcup\limits_{i=1}^n\{z\in C:|z-a_{ii}|\leq \sum\limits_{j=1,j\neq i}^n|a_{ij}|\}$ . For some special matrices, the region can be confined to $\bigcup\limits_{i=1}^n\{z\in C:|z-a_{ii}|\leq \sum\limits_{j=1,j\neq i}^n|a_{ij}|\}\backslash\{z\in C:|z-a_{kk}|<\sum\limits_{j=1,j\neq k}^n|a_{kj}|\}$ for some $k$ . I wonder if the new region above is valid in general?

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Posted by Miwa Lin
updated November 18th, 2009

3 comments

Algebra

trace inequality ★★

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Let $A,B$ be positive semidefinite, by Jensen's inequality, it is easy to see $[tr(A^s+B^s)]^{\frac{1}{s}}\leq [tr(A^r+B^r)]^{\frac{1}{r}}$ , whenever $s>r>0$ .

What about the $tr(A^s+B^s)^{\frac{1}{s}}\leq tr(A^r+B^r)^{\frac{1}{r}}$ , is it still valid?

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Posted by Miwa Lin
updated December 20th, 2010

3 comments

empty triangle

general position

Geometry

Monochromatic empty triangles ★★★

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If $X \subseteq {\mathbb R}^2$ is a finite set of points which is 2-colored, an empty triangle is a set $T \subseteq X$ with $|T|=3$ so that the convex hull of $T$ is disjoint from $X \setminus T$ . We say that $T$ is monochromatic if all points in $T$ are the same color.

Conjecture There exists a fixed constant $c$ with the following property. If $X \subseteq {\mathbb R}^2$ is a set of $n$ points in general position which is 2-colored, then it has $\ge cn^2$ monochromatic empty triangles.