osmootsets

Osmootsets are a class of subsets of a metric space that are defined through a smoothing operation applied to their indicator functions. Let (X, d) be a metric space and let K_t be a smoothing kernel with nonnegative values that integrates to one, parameterized by a scale t > 0. For a subset O ⊆ X, define the smoothed membership function f_t(x) = ∫_X K_t(x, y) 1_O(y) dy. A subset O is called an osmootset if there exists a scale t > 0 and a threshold θ ∈ (0, 1) such that f_t(x) > θ for every x in O and f_t(x) < θ for every x not in O. Equivalently, O is the level set corresponding to a smoothed indicator function, which balances fidelity to the original boundary with robustness to small perturbations.

With a fixed kernel, increasing the scale t generally yields greater boundary blurring, while decreasing t makes the osmootset resemble the original set more closely. For common kernels such as the Gaussian kernel, many familiar sets can be realized as osmootsets by choosing appropriate t and θ. The concept provides a practical bridge between strict set definitions and robust, data-friendly representations.

Properties and relations: The family of osmootsets can often be stabilized under standard set operations. Under suitable choices of the kernel and parameters, unions and intersections of osmootsets can yield new osmootsets, contributing to a lattice-like structure. The construction also induces a topology on X, sometimes referred to as the osmoot topology, that reflects robustness to boundary perturbations rather than exact membership.

Applications and origin: Osmootsets arise in computational topology, image analysis, and geometric data representation as a robust alternative to sharp boundary definitions. They facilitate stable shape analysis, segmentation, and clustering where exact boundaries are unreliable or noisy. The notion is rooted in smoothing and level-set ideas common in applied mathematics.