Convex Optimization · Problem 2.11

Convex functions — Boyd & Vandenberghe

Mathematics — Optimization · hyperbolic-sets

Problem

Hyperbolic sets. Show that the hyperbolic set {x∈R+2∣x1x2≥1}\{x \in \mathbf{R}^2_+ \mid x_1 x_2 \geq 1\} is convex. As a generalization, show that {x∈R+n∣∏i=1nxi≥1}\{x \in \mathbf{R}^n_+ \mid \prod_{i=1}^n x_i \geq 1\} is convex. Hint. If a,b≥0a, b \geq 0 and 0≤θ≤10 \leq \theta \leq 1, then aθb1−θ≤θa+(1−θ)ba^\theta b^{1-\theta} \leq \theta a + (1-\theta)b; see \S3.1.9.

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