Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

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

Duration: 30 minutes

Location: OAT S15/S16/S17

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.

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