New Estimates of Numerical Values Related to a Simplex

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. %Numerical experiments allow to discover Another new result is an inequality \(\xi_6\ <6.0166\). %Прежняя оценка имела вид \(6\leq \xi_6\leq 6.6\). 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~\(C(Q_n)\).

simplex, cube, homothety, axial diameter, interpolation, projection, numerical methods

1. Klimov V. S., Ukhalov A. Yu., Reshenie zadach matematicheskogo analiza s ispolzovaniem sistem kompyuternoi matematiki, Yaroslavl: P. G. Demidov Yaroslavl State University, 2014, (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.

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).

6. Nevskii M. V., Geometricheskie ocenki v polinomialnoy interpolyacii, Yaroslavl: P. G. Demidov Yaroslavl State University, 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.

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.

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.

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

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.

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

15. Scott P. R., “Properties of axial diameters”, Bull. Austral. Math. Soc., 39 (1989), 329–333.