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On directional Whitney inequality

Part of: Approximations and expansions General convexity

Published online by Cambridge University Press:  26 February 2021

Feng Dai  and
Andriy Prymak
Feng Dai*
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, ABT6G 2G1, Canada
Andriy Prymak
Affiliation:
Department of Mathematics, University of Manitoba, Winnipeg, MBR3T 2N2, Canada e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

This paper studies a new Whitney type inequality on a compact domain $\Omega \subset {\mathbb R}^d$ that takes the form

$$ \begin{align*} \inf_{Q\in \Pi_{r-1}^d(\mathcal{E})} \|f-Q\|_p \leq C(p,r,\Omega) \omega_{\mathcal{E}}^r(f,\mathrm{diam}(\Omega))_p,\ \ r\in {\mathbb N},\ \ 0<p\leq \infty, \end{align*} $$
where $\omega _{\mathcal {E}}^r(f, t)_p$ denotes the rth order directional modulus of smoothness of $f\in L^p(\Omega )$ along a finite set of directions $\mathcal {E}\subset \mathbb {S}^{d-1}$ such that $\mathrm {span}(\mathcal {E})={\mathbb R}^d$ , $\Pi _{r-1}^d(\mathcal {E}):=\{g\in C(\Omega ):\ \omega ^r_{\mathcal {E}} (g, \mathrm {diam} (\Omega ))_p=0\}$ . We prove that there does not exist a universal finite set of directions $\mathcal {E}$ for which this inequality holds on every convex body $\Omega \subset {\mathbb R}^d$ , but for every connected $C^2$ -domain $\Omega \subset {\mathbb R}^d$ , one can choose $\mathcal {E}$ to be an arbitrary set of d independent directions. We also study the smallest number $\mathcal {N}_d(\Omega )\in {\mathbb N}$ for which there exists a set of $\mathcal {N}_d(\Omega )$ directions $\mathcal {E}$ such that $\mathrm {span}(\mathcal {E})={\mathbb R}^d$ and the directional Whitney inequality holds on $\Omega $ for all $r\in {\mathbb N}$ and $p>0$ . It is proved that $\mathcal {N}_d(\Omega )=d$ for every connected $C^2$ -domain $\Omega \subset {\mathbb R}^d$ , for $d=2$ and every planar convex body $\Omega \subset {\mathbb R}^2$ , and for $d\ge 3$ and every almost smooth convex body $\Omega \subset {\mathbb R}^d$ . For $d\ge 3$ and a more general convex body $\Omega \subset {\mathbb R}^d$ , we connect $\mathcal {N}_d(\Omega )$ with a problem in convex geometry on the X-ray number of $\Omega $ , proving that if $\Omega $ is X-rayed by a finite set of directions $\mathcal {E}\subset \mathbb {S}^{d-1}$ , then $\mathcal {E}$ admits the directional Whitney inequality on $\Omega $ for all $r\in {\mathbb N}$ and $0<p\leq \infty $ . Such a connection allows us to deduce certain quantitative estimate of $\mathcal {N}_d(\Omega )$ for $d\ge 3$ .

A slight modification of the proof of the usual Whitney inequality in literature also yields a directional Whitney inequality on each convex body $\Omega \subset {\mathbb R}^d$ , but with the set $\mathcal {E}$ containing more than $(c d)^{d-1}$ directions. In this paper, we develop a new and simpler method to prove the directional Whitney inequality on more general, possibly nonconvex domains requiring significantly fewer directions in the directional moduli.

Keywords

Whitney-type inequalitydirectional modulus of smoothnessconvex domainsC2-domainsmultivariate polynomialsillumination of convex bodiesX-ray numberillumination number

MSC classification

Primary: 41A10: Approximation by polynomials
Secondary: 41A25: Rate of convergence, degree of approximation 41A63: Multidimensional problems (should also be assigned at least one other classification number in this section) 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 52A40: Inequalities and extremum problems
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 3 , June 2022 , pp. 833 - 857
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

The first author was supported by NSERC of Canada Discovery grant RGPIN-2020-03909, and the second author was supported by NSERC of Canada Discovery grant RGPIN-2020-05357.

References

Besicovitch, A. S., Measure of asymmetry of convex curves. J. Lond. Math. Soc. 23(1948), 237–240.CrossRefGoogle Scholar
Bezdek, K. and Khan, M. A., The geometry of homothetic covering and illumination . In: Discrete geometry and symmetry, Springer Proc. Math. Stat., 234, Springer, Cham, Switzerland, 2018, pp. 1–30.Google Scholar
Bezdek, K. and Kiss, Gy., On the X-ray number of almost smooth convex bodies and of convex bodies of constant width. Canad. Math. Bull. 52(2009), no. 3, 342–348.CrossRefGoogle Scholar
Bondarenko, A. V., Prymak, A., and Radchenko, D., On concentrators and related approximation constants. J. Math. Anal. Appl. 402(2013), no. 1, 234–241.CrossRefGoogle Scholar
Brenner, S. C. and Scott, L. R., The mathematical theory of finite element methods. Texts in Applied Mathematics, 15, Springer-Verlag, New York, (1994).CrossRefGoogle Scholar
Brudnyi, J. A., A multidimensional analogue of a certain theorem of Whitney. Mat. Sb. (N.S.) 82(1970), no. 124, 175–191 (Russian).Google Scholar
Brudnyi, Y. A., On a theorem of local best approximations. Kazan. Gos. Univ. Uchen. Zap. 124(1964), no. kn. 6, 43–49 (Russian).Google Scholar
Brudnyi, Y. A. and Kalton, N. J., Polynomial approximation on convex subsets of R n . Constr. Approx. 16(2000), no. 2, 161–199.CrossRefGoogle Scholar
Cohen, M. B., Cousins, B., Lee, Y. T., and Yang, X., A near-optimal algorithm for approximating the John ellipsoid. In: Proceedings of the 32nd Conference on Learning Theory, 2019, pp. 849–873. http://proceedings.mlr.press/v99/cohen19a.html.Google Scholar
Damla Ahipasaoglu, S., Sun, P., and Todd, M. J., Linear convergence of a modified Frank–Wolfe algorithm for computing minimum-volume enclosing ellipsoids. Optim. Methods Softw. 23(2008), no. 1, 5–19.CrossRefGoogle Scholar
Dekel, S. and Leviatan, D., Whitney estimates for convex domains with applications to multivariate piecewise polynomial approximation. Found. Comput. Math. 4(2004), no. 4, 345–368.CrossRefGoogle Scholar
DeVore, R. A. and Lorentz, G. G., Constructive approximation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 303, Springer-Verlag, Berlin, Germany, 1993.CrossRefGoogle Scholar
Ditzian, Z., Polynomial approximation in ${L}_p(S)$ for $p>0$ . Constr. Approx. 12(1996), no. 2, 241–269.Google Scholar
Ditzian, Z. and Ivanov, K. G., Minimal number of significant directional moduli of smoothness. Anal. Math. 19(1993), no. 1, 13–27 (English, with Russian summary).CrossRefGoogle Scholar
Ditzian, Z. and Prymak, A., Ul $^{\prime }$ yanov-type inequality for bounded convex sets in R d . J. Approx. Theory 151(2008), no. 1, 60–85.CrossRefGoogle Scholar
Dung, D. and Ullrich, T., Whitney type inequalities for local anisotropic polynomial approximation. J. Approx. Theory 163(2011), no. 11, 1590–1605.CrossRefGoogle Scholar
Gilewicz, J., Kryakin, Y. V., and Shevchuk, I. A., Boundedness by 3 of the Whitney interpolation constant. J. Approx. Theory 119(2002), no. 2, 271–290.CrossRefGoogle Scholar
Hammer, P. C., Approximation of convex surfaces by algebraic surfaces. Mathematika 10(1963), 64–71.CrossRefGoogle Scholar
Huang, H., Slomka, B. A., Tkocz, T., and Vritsiou, B.-H., Improved bounds for Hadwiger’s covering problem via thin-shell estimates. Preprint, 2018. arxiv:1811.12548 Google Scholar
Iurchenko, I., Properties of extremal convex bodies. M.Sc. thesis, University of Manitoba, 2012. http://hdl.handle.net/1993/9155.Google Scholar
Ivanov, K. G. and Takev, M. D., $O\left(n\ln n\right)$ bound for Whitney constants. C. R. Acad. Bulg. Sci. 38(1985), no. 9, 1129–1131.Google Scholar
John, F., Extremum problems with inequalities as subsidiary conditions . In: Studies and essays presented to R. Courant on his 60th birthday, January 8, 1948, Interscience Publishers, Inc., New York, 1948, pp. 187–204.Google Scholar
Papadoperakis, I., An estimate for the problem of illumination of the boundary of a convex body in E3 . Geom. Dedicata. 75(1999), no. 3, 275–285.CrossRefGoogle Scholar
Prymak, A. and Shepelska, V., On illumination of the boundary of a convex body in ${E}^n$ , $n=4,5,6$ . Preprint, 2018. arxiv:1811.08962 Google Scholar
Rogers, C. A., A note on coverings. Mathematika 4(1957), 1–6.CrossRefGoogle Scholar
Rogers, C. A. and Shephard, G. C., The difference body of a convex body. Arch. Math. (Basel) 8(1957), 220–233.CrossRefGoogle Scholar
Sendov, Bl., The constants of H. Whitney are bounded. C. R. Acad. Bulgare Sci. 38(1985), no. 10, 1299–1302.Google Scholar
Storozhenko, E. A., Approximation by algebraic polynomials of functions of the class ${L}^p$ , $0<p<1$ . Izv. Akad. Nauk SSSR Ser. Mat. 41(1977), no. 3, 652–662, 718 (Russian).Google Scholar
Sun, P. and Freund, R. M., Computation of minimum-volume covering ellipsoids. Oper. Res. 52(2004), no. 5, 690–706.CrossRefGoogle Scholar
Whitney, H., On functions with bounded $n$ th differences. J. Math. Pures Appl. (9) 36(1957), 67–95.Google Scholar