13.1 Convex hypersurfaces and their properties

Convex hypersurfaces are fascinating geometric objects in n-dimensional Euclidean space. They're defined by staying on one side of their tangent hyperplanes and have unique properties like being closed, bounded, and separating space into interior and exterior regions.

The second fundamental form is key to understanding convexity. It measures curvature and must be positive semidefinite for a hypersurface to be convex. This condition ensures non-negative principal curvatures, maintaining the hypersurface's overall shape consistency.

Foundations of Convex Hypersurfaces

Properties of convex hypersurfaces

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• Convex hypersurface defined as $(n-1)$-dimensional submanifold of $n$-dimensional Euclidean space locally lies on one side of its tangent hyperplane at each point
• Key properties encompass closed and bounded nature separates space into two disjoint regions (interior and exterior) supports unique tangent hyperplane at each point
• Examples include sphere ellipsoid paraboloid demonstrating diverse geometric forms
• Gauss map associates each point on hypersurface with its unit normal vector proves injective for convex hypersurfaces enabling unique identification of points

Convexity and second fundamental form

• Second fundamental form measures curvature of hypersurface represented by symmetric matrix of shape operator
• Positive semidefinite condition requires all eigenvalues to be non-negative equivalent to non-negative principal curvatures
• Proof outline demonstrates:
  1. Convexity implies positive semidefinite second fundamental form
  2. Positive semidefinite second fundamental form implies convexity
• Geometric interpretation reveals positive semidefinite second fundamental form ensures local convexity at each point maintaining overall shape consistency

Curvature and Geometric Implications

Convexity vs principal curvatures

• Principal curvatures as eigenvalues of shape operator measure maximum and minimum curvatures at a point
• Convexity condition necessitates all principal curvatures to be non-negative
• Special cases include strictly convex (all principal curvatures positive) and flat regions (at least one principal curvature zero)
• Gaussian curvature calculated as product of principal curvatures always non-negative for convex hypersurfaces
• Mean curvature averaged from principal curvatures remains non-negative for convex hypersurfaces indicating overall positive curvature

Geometric implications of convexity

• Shape characteristics preclude self-intersections and saddle points or hyperbolic regions maintaining smooth contours
• Convexity preserved under affine transformations (scaling, rotation, translation) intersection of convex hypersurfaces remains convex
• Support function uniquely represents convex hypersurface determines shape and properties enabling analytical study
• Applications extend to optimization (convex programming constraints) and differential geometry (minimal surfaces constant mean curvature surfaces)
• Duality concepts explore polar sets and dual hypersurfaces establishing relationship between convex hypersurfaces and their duals enhancing geometric understanding