Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

Mittagsseminar Talk Information

Date and Time: Tuesday, March 03, 2015, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Yasuyuki Tsukamoto (Kyoto University)

Objects with projection images just like a cube

This is a joint work with Hideki Tsuiki. Imaginary cubes are objects which have square projection images in three orthogonal ways just like a cube has. The combinatorial types of imaginary cubes have been studied, and we make generalizations about imaginary cubes in higher-dimensional cases.

Let $d$ be a positive integer no less than three. We consider compact subsets of $d$-dimensional Euclidean space, and we call them ``objects''. Let $C$ be a $d$-dimensional hypercube ($d$-cube for short) and $p_1,\ldots,p_d$ be projections along the edges of $C$. We say that an object $A$ is an imaginary cube of $C$ if it satisfies $p_i(A)=p_i(C)$ for $i=1,\ldots,d$. The main result is the following.

Suppose that an object $A$ is an imaginary cube of two different $d$-cubes $C$ and $C'$. Then (i) $C$ and $C'$ share one of their diagonals, (ii) $d$ is no greater than four, (iii) if $A$ is convex in addition, then $A$ is the intersection of $C$ and $C'$. In four-dimensional case, actually, $C$ and $C'$ share four of their diagonals, and their intersection is a 16-cell.

The three-dimensional case of this theorem is already known. We give a proof applicable to higher-dimensional cases.

Upcoming talks     |     All previous talks     |     Talks by speaker     |     Upcoming talks in iCal format (beta version!)

Previous talks by year:   2018  2017  2016  2015  2014  2013  2012  2011  2010  2009  2008  2007  2006  2005  2004  2003  2002  2001  2000  1999  1998  1997  1996  

Information for students and suggested topics for student talks

Automatic MiSe System Software Version 1.4803M   |   admin login