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lcrp

The radial projection, $\rho$, on a normed space $X$ is the (nonlinear) mapping given by $$ \rho(x) = \begin{cases} x/\left\|x\right\| & \text{for $\left\|x\right\| \geq 1$} \\ x & \text{otherwise.} \end{cases} $$ The Lipschitz constant for this mapping, the minimal $k$ such that $$ \left\| \rho(x) - \rho(y) \right| \leq k \left\| x-y \right\| \qquad(x,\,y\in X), $$ is a quantity of interest in the geometry of normed and Banach spaces. In a recent paper it is determined that the Lipschitz constant when $X$ is the sequence space $\ell_p$ is given by the maximum of a particular one-dimensional $p$-dependent function. This has an application in the efficient calculation of the $(p,q)$-operator norms of matrices, i.e., the norm of a matrix considered as an operator $\ell_p\rightarrow\ell_q$. See, for example, the opnorm package.

Below we provide some code to efficiently evaluate this Lipschitz constant.

• lcrp.m for Octave/Matlab
• the C99/C++ implementation
• a technical note on the numerical aspects of the calculation.