TABLE OF CONTENTS Part I Local Conditions for Containment Chapter Page 0. INTRODUCTION 1 1. MAIN RESULT AND SKETCH PROOF 2 1.1 Definitions and Statement of Main Result 2 2. THE MAIN RESULT FOR CURVES 6 2.1 Preliminary Results and Notation 6 2.2 The Main Result for Curves 7 3. CONVEX REGIONS IN R n 21 3.1 Preliminary Results 21 3.2 Faithful Projections 23 3.3 Proving Lemma 1.1.5 26 3.4 Proving Theorem 1.1.4 33 3.5 Possible Generalisations of the Main Theorem 41 4. THE SMOOTH CASE: APPLICATIONS TO DIFFERENTIAL GEOMETRY 47 4.1 Local Representation of 5 as a Function 47 4.2 Radii of Curvature Indicatrices 48 4.2 Semi-Local Insideness in Terms of Radii of Curvature and Indicatrices 50 continued over page iii
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