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I want to find an $n$-vector $\hat{\theta}$ that maximizes a function $f(\theta)$ subject to the $p$-norm constraint $||\theta||_p = c$.

Is there a general parametrization of $p$-norm hyperspheres that makes this easy?

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It depends on the value of $p$: For $p=1$ and $p=\infty$ the unit sphere is a polyhedron, for $1<p<\infty$ a smooth surface. – Christian Blatter Mar 10 '11 at 9:24
up vote 2 down vote accepted

The standard 2-norm hypersphere is parametrized by hyperspherical coordinates, which you can easily turn into a parametrization of the $p$-norm hypersphere by transforming each coordinate as $x_i \mapsto \operatorname{sgn}(x_i) \lvert x_i \rvert^{2/p}$.

However, I would not recommend using this for solving an optimization problem, because the parametrization is singular at the poles, and at many other points when $p \le 1$ or $p = \infty$. You should look into Lagrange multipliers instead.

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