المؤلفون: A. Yu. Ukhalov, Mikhail Nevskii

المصدر: Automatic Control and Computer Sciences. 52:667-679

مصطلحات موضوعية: Simplex, N dimensional, Computer science, Image (category theory), 010102 general mathematics, Center (category theory), Order (ring theory), 02 engineering and technology, 01 natural sciences, Combinatorics, Control and Systems Engineering, Signal Processing, 0202 electrical engineering, electronic engineering, information engineering, 020201 artificial intelligence & image processing, 0101 mathematics, Software

الوصف: Let $$n \in \mathbb{N}$$ , $${{Q}_{n}} = {{[0,1]}^{n}}.$$ For a nondegenerate simplex $$S \subset {{\mathbb{R}}^{n}}$$ , by $$\sigma S$$ we denote the homothetic image of $$S$$ with center of homothety in the center of gravity of $$S$$ and ratio of homothety $$\sigma $$ . By $${{d}_{i}}(S)$$ we mean the $$i$$ th axial diameter of $$S$$ , i. e. the maximum length of a line segment in $$S$$ parallel to the $$i$$ th coordinate axis. Let $$\xi (S) = min{\text{\{ }}\sigma \geqslant 1:{{Q}_{n}} \subset \sigma S{\text{\} }},$$ $${{\xi }_{n}} = min{\text{\{ }}\xi (S):S \subset {{Q}_{n}}{\text{\} }}.$$ By $$\alpha (S)$$ we denote the minimal $$\sigma > 0$$ such that $${{Q}_{n}}$$ is contained in a translate of simplex $$\sigma S$$ . Consider $$(n + 1) \times (n + 1)$$ -matrix $${\mathbf{A}}$$ with the rows containing coordinates of vertices of $$S$$ ; last column of $${\mathbf{A}}$$ consists of 1’s. Put $${{{\mathbf{A}}}^{{ - 1}}}$$ $$ = ({{l}_{{ij}}})$$ . Denote by $${{\lambda }_{j}}$$ linear function on $${{\mathbb{R}}^{n}}$$ with coefficients from the $$j$$ th column of $${{{\mathbf{A}}}^{{ - 1}}}$$ , i.e. $${{\lambda }_{j}}(x) = {{l}_{{1j}}}{{x}_{1}} + \ldots + {{l}_{{nj}}}{{x}_{n}} + {{l}_{{n + 1,j}}}.$$ Earlier the first author proved the equalities $$\tfrac{1}{{{{d}_{i}}(S)}} = \tfrac{1}{2}\sum\nolimits_{j = 1}^{n + 1} \left| {{{l}_{{ij}}}} \right|,\alpha (S) = \sum\nolimits_{i = 1}^n \tfrac{1}{{{{d}_{i}}(S)}}.$$ In the present paper we consider the case $$S \subset {{Q}_{n}}$$ . Then all the $${{d}_{i}}(S) \leqslant 1$$ , therefore, $$n \leqslant \alpha (S) \leqslant \xi (S).$$ If for some simplex $$S{\kern 1pt} {{'}} \subset {{Q}_{n}}$$ holds $$\xi (S{\kern 1pt} {{'}}) = n,$$ then $${{\xi }_{n}} = n$$ , $$\xi (S{\kern 1pt} {{'}}) = \alpha (S{\kern 1pt} {{'}})$$ , and $${{d}_{i}}(S{\kern 1pt} {{'}}) = 1$$ . However, such the simplices S ' exist not for all the dimensions $$n$$ . The first value of $$n$$ with such a property is equal to $$2$$ . For each 2-dimensional simplex, $$\xi (S) \geqslant {{\xi }_{2}} = 1 + \tfrac{{3\sqrt 5 }}{5} = 2.34 \ldots > 2$$ . We have an estimate $$n \leqslant {{\xi }_{n}} < n + 1$$ . The equality $${{\xi }_{n}} = n$$ takes place if there exist an Hadamard matrix of order $$n + 1$$ . Further investigation showed that $${{\xi }_{n}} = n$$ also for some other $$n$$ . In particular, simplices with the condition $$S \subset {{Q}_{n}} \subset nS$$ were built for any odd $$n$$ in the interval $$1 \leqslant n \leqslant 11$$ . In the first part of the paper we present some new results concerning simplices with such a condition. If $$S \subset {{Q}_{n}} \subset nS$$ , then center of gravity of $$S$$ coincide with center of $${{Q}_{n}}$$ . We prove that $$\sum\nolimits_{j = 1}^{n + 1} \left| {{{l}_{{ij}}}} \right| = 2\,(1 \leqslant i \leqslant n),\sum\nolimits_{i = 1}^n \left| {{{l}_{{ij}}}} \right| = \tfrac{{2n}}{{n + 1}}(1 \leqslant j \leqslant n + 1).$$ Also we give some corollaries. In the second part of the paper we consider the following conjecture. Let for simplex $$S \subset {{Q}_{n}}$$ an equality $$\xi (S) = {{\xi }_{n}}$$ holds. Then $$(n - 1)$$ -dimensional hyperplanes containing the faces of $$S$$ cut off from the cube $${{Q}_{n}}$$ the equal-sized parts. Though it is true for $$n = 2$$ and $$n = 3$$ , in general case this conjecture is not valid.

URL الوصول: https://explore.openaire.eu/search/publication?articleId=doi_________::f9d181e6c8e127494fb57b8400ea5503
https://doi.org/10.3103/s0146411618070192