For $n\ge 1$, every Euclidean closed ball and every Euclidean sphere of positive radius is compact

Depends on

• Euclidean spheres and closed balls as subspaces of $\mathbb{R}^n$
• For a nonempty subset of $\mathbb{R}^n$ with $n\ge1$, compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent
• The finite and reverse triangle inequalities for a norm; and for $n \ge 1$ every norm $N$ on $\mathbb{R}^n$ satisfies $N(x) \le C\lVert x\rVert_1$ and is Lipschitz, hence continuous, for $d_2$
• The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement
• Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space

Used by

• Ball Poisson integrals converge uniformly along radial boundary approaches Corollary
• ℝⁿ is locally compact and σ-compact Corollary
• Sphere self-maps are homotopic exactly when their degrees agree Corollary
• The constructed classical solutions are locally determined by the Cauchy data Corollary
• Outward dilation defeats subcritical inclusion and homogeneous Poincare on Euclidean space Counterexample
• Poincare-Wirtinger fails on disconnected bounded domains Counterexample
• The W^1,p→ L^q bound fails for q>p^* by dilation Counterexample
• John domains and the John constant Definition
• Spherical means and the weighted ball integral of space-dependent data Definition
• A circle is the boundary of a disk Example
• A point source produces a uniform expanding sphere Example
• Compactness and Liouville distinguish the three models Example
• Maps from even projective space to the sphere use mod-two degree Example
• Newton shell theorem from harmonic mean values Example
• Signed points give the oriented zero-bordism invariant Example
• The Euclidean closed ball and sphere worked through the compactness equivalence chart Example
• The pair of pants is a cobordism realizing addition of circles Example
• Two unoriented points bound an interval Example
• Bishop gromov volume ratio is nondecreasing under a ricci lower bound False statement
• The laplace beltrami definition licenses the use of all euclidean harmonic function theory on manifolds False statement
• A compact free affine plane quotient comes from a rank-two lattice Lemma
• Ball means and sphere means are related by a radial derivative Lemma
• Cap and complement estimate for the ball Poisson integral Lemma
• Coordinate balls form a basis of a topological manifold Lemma
• Cylinders give reflexivity of cobordism Lemma
• Differentiating an integral with moving endpoints Lemma
• Every integer is realized by a map to the sphere Lemma
• Failure of the supersolution test for the lower envelope allows a local bump Lemma
• Finiteness, superlinearity of the Lagrangian and localisation of Hopf--Lax near-minimisers Lemma
• Global cubic minimizers, admissible inexact steps, and cubic Cauchy decrease Lemma
• Global geodesic-polar coordinates from a boundary point on a simple manifold Lemma
• Mean-zero Poincare estimate on bounded connected extension domains below the dimension Lemma
• Polynomial traces, monomial basis and bounded evaluation for the ball Hardy space Lemma
• Relatively compact coordinate balls and half-balls form a boundary-manifold basis Lemma
• Removing a compact chart disc gives a Greenian surface Lemma
• Riemannian Green identity, boundary trace, and wave lifting Lemma
• Smoothness of the Bergman kernel and positivity of its diagonal on bounded domains Lemma
• Smoothness, parity and zero-radius limits of spherical means Lemma
• Sphere integrals of a cylindrical function project to weighted ball integrals Lemma
• Spherical harmonic modes are orthogonal, complete for continuous data, and have polynomial evaluation bounds Lemma

…and 27 more results.

Dependency tree · two levels

47 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Heine-Borel theorem (standard reference, not scraped)