Convex Surfaces

Abstract

An oriented n-surface S in ℝ^{n +1} is convex (or globally convex) if, for each p ∈ S, S is contained in one of the closed half-spaces

$$H_p^ + \, = \,\left\{ {q\, \in \,{\mathbb{R}^{n\, + \,1}}:\,\left( {q\, - \,p} \right)\, \cdot \,N\left( p \right)\, \geqslant \,0} \right\}$$

or

$$H_p^ - \, = \,\left\{ {q\, \in \,{\mathbb{R}^{n\, + \,1}}:\,\left( {q\, - \,p} \right)\, \cdot \,N\left( p \right)\, \leqslant \,0} \right\}$$

where N is the Gauss map of S (see Figure 13.1). An oriented n-surface S is convex at p ∈ S if there exists an open set V ⊂ ℝ^{n +1} containing p such that S ∩ V is contained either in H ⁺ _p or in H ⁻ _p . Thus a convex n-surface is necessarily convex at each of its points, but an n-surface convex at each point need not be a convex n-surface (see Figure 13.2).