Convex Geometry & Measure Constructions

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Abstractions about convexity, covering, and measure in geometric analysis, including convex function and fixed-point results (proper convex function, Schauder fixed-point theorem), covering and directional geometric problems (Kakeya set, Tarski's plank problem, edge tessellation), and space-comparison or construction tools (Gromov-Hausdorff convergence, box spline, spherical design).

12 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Balayage — An operator that sweeps an interior measure onto a domain boundary while preserving its Newtonian potential outside the closed domain.
• Box Spline — A compactly supported multivariate piecewise-polynomial function generated from a finite multiset of direction vectors, equivalently by repeated convolution of uniform segment measures or projection of a higher-dimensional box.
• Centerpoint (Geometry) — Choose a point that every containing closed halfspace shares with at least ⌈n/(d+1)⌉ of n points in d dimensions.
• Edge Tessellation — A congruent polygonal tiling closed under reflection across every tile edge, so one tile and its edge reflections generate the entire tiling.
• Euclidean Vector — A free Euclidean vector is a location-independent class of directed segments with linear operations and Euclidean length and angle.
• Gromov–Hausdorff convergence — A convergence notion for metric spaces defined by their Gromov–Hausdorff distance tending to zero after comparison in common ambient metrics.
• John Ellipsoid — The unique maximum-volume ellipsoid inscribed in a full-dimensional compact convex body, providing a centered geometric approximation whose bound depends on symmetry.
• Kakeya Set — A subset of Euclidean space containing a unit line segment in every direction, whose directional coverage can coexist with vanishing measure and extreme geometric overlap.
• Proper Convex Function — An extended-real convex function whose effective domain is nonempty and which nowhere takes negative infinity, excluding the two degenerate functions that break convex-analytic operations.
• Schauder Fixed-Point Theorem — A continuous self-map of a nonempty closed convex set has a fixed point when its image is relatively compact in the surrounding locally convex space.
• Spherical Design — A finite equal-weight point set on a unit sphere whose discrete average exactly matches the sphere average for every polynomial through a declared degree.
• Tarski's Plank Problem — Relate geometric coverage to directional thickness: any finite family of Euclidean planks covering a convex body must have total plank width at least the body's minimum width.