AMBIENTUM BIOETHICA BIOLOGIA CHEMIA DIGITALIA DRAMATICA EDUCATIO ARTIS GYMNAST. ENGINEERING EPHEMERIDES EUROPAEA GEOGRAPHIA GEOLOGIA HISTORIA HISTORIA ARTIUM INFORMATICA IURISPRUDENTIA MATHEMATICA MUSICA NEGOTIA OECONOMICA PHILOLOGIA PHILOSOPHIA PHYSICA POLITICA PSYCHOLOGIA-PAEDAGOGIA SOCIOLOGIA THEOLOGIA CATHOLICA THEOLOGIA CATHOLICA LATIN THEOLOGIA GR.-CATH. VARAD THEOLOGIA ORTHODOXA THEOLOGIA REF. TRANSYLVAN
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STUDIA MATHEMATICA - Ediţia nr.2 din 2022 | |||||||
Articol: |
THE LEVEL SETS OF FUNCTIONS WITH BOUNDED CRITICAL SETS AND BOUNDED HESS+ COMPLEMENTS. Autori: CORNEL PINTEA. |
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Rezumat: DOI: 10.24193/subbmath. 2022.2.18 Published Online: 2022-06-10 Published Print: 2022-06-30 pp. 441-454 VIEW PDF FULL PDF Abstract: We denote by ${ m Hess}^+(f)$ the set of all points $pinmathbb{R}^n$ such that the Hessian matrix $H_p(f)$ of the $C^2$-smooth function $f:mathbb{R}^nlongrightarrowmathbb{R}$ is positive definite. In this paper we prove several properties of real-valued functions of several variables by showing the connectedness of their level sets for sufficiently high levels, under the boundedness assumption on the critical set. In the case of three variables we also prove the convexity of the levels surfaces for sufficiently high levels, under the additional boundedness assumption on the ${ m Hess}^+$ complement. The selection of the {em a priori} convex levels, among the connected regular ones, is done through the positivity of the Gauss curvature function which ensure an ovaloidal shape of the levels to be selected. The ovaloidal shape of a level set makes a diffeomorphism out of the associated Gauss map. This outcome Gauss map diffeomorphism is then extended to a smooth homeomorphism which is used afterwards to construct one-parameter families of smooth homeomorphisms of Loewner chain flavor. Key words: The Hessian matrix, The ${ m Hess}^+$ region, curvature, Gauss curvature, convex curves, ovaloids Mathematics Subject Classification (2010): 14Q10, 52A10, 52A15, 53A05. |
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