New estimates of numerical values related to a simplex

M. V. Nevskii ; A. Yu. Ukhalov

Modelirovanie i analiz informacionnyh sistem, Tome 24 (2017) no. 1, pp. 94-110.

Résumé

Let $n\in {\mathbb N}$ and $Q_n=[0,1]^n$. For a nondegenerate simplex $S\subset {\mathbb R}^n$, by $\sigma S$ we denote the homothetic copy of $S$ with center of homothety in the center of gravity of $S$ and ratio of homothety $\sigma$. By $\xi(S)$ we mean the minimal $\sigma>0$ such that $Q_n\subset \sigma S$. By $\alpha(S)$ denote the minimal $\sigma>0$ such that $Q_n$ is contained in a translate of $\sigma S$. By $d_i(S)$ we denote the $i$th axial diameter of $S$, i. e. the maximum length of the segment contained in $S$ and parallel to the $i$th coordinate axis. Formulae for $\xi(S)$, $\alpha(S)$, $d_i(S)$ were proved earlier by the first author. Define $\xi_n=\min\{ \xi(S): S\subset Q_n\}. $ We always have $\xi_n\geq n.$ We discuss some conjectures formulated in the previous papers. One of these conjectures is the following. For every $n$, there exists $\gamma>0$, not depending on $S\subset Q_n$, such that an inequality $\xi(S)-\alpha(S)\leq \gamma (\xi(S)-\xi_n)$ holds. Denote by $\varkappa_n$ the minimal $\gamma$ with such a property. We prove that $\varkappa_1=\frac{1}{2}$; for $n>1$, we obtain $\varkappa_n\geq 1$. If $n>1$ and $\xi_n=n,$ then $\varkappa_n=1$. The equality $\xi_n=n$ holds if $n+1$ is an Hadamard number, i. e. there exists an Hadamard matrix of order $n+1$. This proposition is known; we give one more proof with the direct use of Hadamard matrices. We prove that $\xi_5=5$. Therefore, there exists $n$ such that $n+1$ is not an Hadamard number and nevertheless $\xi_n=n$. The minimal $n$ with such a property is equal to $5$. This involves $\varkappa_5=1$ and also disproves the following previous conjecture of the first author concerning the characterization of Hadamard numbers in terms of homothety of simplices: $n+1$ is an Hadamard number if and only if $\xi_n=n$. This statement is valid only in one direction. There exists a simplex $S\subset Q_5$ such that the boundary of the simplex $5S$ contains all the vertices of the cube $Q_5$. We describe a one-parameter family of simplices contained in $Q_5$ with the property $\alpha(S)=\xi(S)=5.$ These simplices were found with the use of numerical and symbolic computations. Another new result is an inequality $\xi_6\ 6.0166$. We also systematize some of our estimates of numbers $\xi_n$, $\theta_n$, $\varkappa_n$ derived by now. The symbol $\theta_n$ denotes the minimal norm of interpolation projection on the space of linear functions of $n$ variables as an operator from $C(Q_n)$ to $C(Q_n)$.

Détail

Mots-clés : simplex, interpolation
Keywords: cube, homothety, axial diameter, projection, numerical methods.

Bibliographie

[1] Klimov V. S., Ukhalov A. Yu., Reshenie zadach matematicheskogo analiza s ispolzovaniem sistem kompyuternoi matematiki, P. G. Demidov Yaroslavl State University, Yaroslavl, 2014, 96 pp. (in Russian)

[2] Nevskij M. V., “Inequalities for the norms of interpolating projections”, Modeling and Analysis of Information Systems, 15:3 (2008), 28–37 (in Russian)

[3] Nevskij M. V., “On a certain relation for the minimal norm of an interpolational projection”, Modeling and Analysis of Information Systems, 16:1 (2009), 24–43 (in Russian)

[4] Nevskii M. V., “On a property of $n$-dimensional simplices”, Math. Notes, 87:4 (2010), 543–555 | DOI | DOI | MR | Zbl

[5] Nevskii M. V., “On geometric charasteristics of an $n$-dimensional simplex”, Modeling and Analysis of Information Systems, 18:2 (2011), 52–64 (in Russian) | MR

[6] Nevskii M. V., Geometricheskie ocenki v polinomialnoy interpolyacii, P. G. Demidov Yaroslavl State University, Yaroslavl, 2012 (in Russian)

[7] Nevskii M. V., “Computation of the longest segment of a given direction in a simplex”, Journal of Math. Sciences, 203:6 (2014), 851–854 | DOI | MR | Zbl

[8] Nevskii M. V., Ukhalov A. Yu., “On numerical charasteristics of a simplex and their estimates”, Modeling and Analysis of Information Systems, 23:5 (2016), 603–619 (in Russian)

[9] Hall M. (Jr.), Combinatorial theory, Blaisdall publishing company, Waltham (Massachusets)–Toronto–London, 1967 (in English) | MR | Zbl

[10] Hudelson M., Klee V., Larman D., “Largest $j$-simplices in $d$-cubes: some relatives of the Hadamard maximum determinant problem”, Linear Algebra Appl., 241–243 (1996), 519–598 | DOI | MR | Zbl

[11] Lassak M., “Parallelotopes of maximum volume in a simplex”, Discrete Comput. Geom., 21 (1999), 449–462 | DOI | MR | Zbl

[12] Mangano S., Mathematica cookbook, O'Reilly Media Inc., Cambridge, 2010

[13] Nevskii M., “Properties of axial diameters of a simplex”, Discrete Comput. Geom., 46:2 (2011), 301–312 | DOI | MR | Zbl

[14] Scott P. R., “Lattices and convex sets in space”, Quart. J. Math. Oxford (2), 36 (1985), 359–362 | DOI | MR | Zbl

[15] Scott P. R., “Properties of axial diameters”, Bull. Austral. Math. Soc., 39 (1989), 329–333 | DOI | MR | Zbl