For a certain application I need to compute a pointwise supremum of this family of gaussian convolutions:

$$\sup_s f(x)\otimes e^{-\frac{x^2}{s^2}}$$

where $f(x),x\in \mathbb{R}^2$ is known and $\otimes$ is a convolution over the entire plane.

Could you point me to a reference that would help with computation of this, er, "evolute" of convolutions?

For a certain application I need to compute a pointwise supremum of this family of gaussian convolutions:

$$\sup_s f(x)\otimes e^{-\frac{x^2}{s^2}}$$

where $f(x),x\in \mathbb{R}^2$ is known and $\otimes$ is a convolution over the entire plane.

Could you point me to a reference that would help with computation of this envelope of parametrized convolutions?