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Solved 50

• Nonexpansive bijections from balls between c₀₀ and c₀ onto balls between c₀₀ and ℓ∞ are signed permutations, wherever the target matches signsProved

  Oct 2026

• A ball between c₀₀ and ℓ∞ that is the nonexpansive bijective image of a ball between c₀₀ and c₀ lies in c₀Proved

  Oct 2026

• A nonexpansive bijection from a ball between c₀₀ and c₀ onto a ball between c₀₀ and ℓ∞ moves each coordinate inward along a signed permutationProved

  Oct 2026

• Normed spaces: a ball whose sphere is covered by finite-dimensional faces is plasticProved

  Oct 2026

• Sphere covered by finite-dimensional faces, normed spaces: every nonexpansive bijection onto the ball is linearProved

  Oct 2026

• A nonexpansive ball bijection that keeps the sphere is homogeneous at every preimage of a point of a finite-dimensional faceProved

  Oct 2026

• A nonexpansive map of a closed bounded set that is isometric on its relative boundary is isometricProved

  Oct 2026

• Two points of a bounded set at equal distances from every point of its relative boundary coincideProved

  Oct 2026

• A nonexpansive ball map that keeps the sphere is isometric between points on whose rays it is homogeneousProved

  Oct 2026

• A ball whose sphere is covered by finite-dimensional faces is plasticProved

  Oct 2026

• Sphere covered by finite-dimensional faces: every nonexpansive bijection onto the ball is linearProved

  Oct 2026

• Every finite-dimensional face of the target ball is carried onto a face of the same dimensionProved

  Oct 2026

• A face whose boundary preimage is relatively compact is carried onto a face of the same dimensionProved

  Oct 2026

• If the boundary preimage of a face is compact, extreme and symmetric, the preimage is a face of the same dimensionProved

  Oct 2026

• The reflected preimage of a face is the cone over the boundary preimage from any interior preimageProved

  Oct 2026

• A compact preimage face is carried onto the face at every scaleProved

  Oct 2026

• Continuous injections between convex sets of equal finite dimension preserve relative interiorsProved

  Oct 2026

• A nonempty compact convex set with finite-dimensional span is a closed ball in its direction spaceProved

  Oct 2026

• If the smaller faces behave, the preimage of the relative boundary of a face is relatively compactProved

  Oct 2026

• A point of a set lies in its relative interior iff a small ball of its direction space around the point stays in the setProved

  Oct 2026

• Invariance of domain between finite-dimensional normed spaces of equal dimensionProved

  Oct 2026

• A convex set mapped continuously and injectively into a finite-dimensional affine subspace has no larger dimensionProved

  Oct 2026

• A continuous self-map of a closed ball fixing its boundary sphere is onto the ballProved

  Oct 2026

• A one-point face of the target ball is carried onto a one-point faceProved

  Oct 2026

• A finite-dimensional face of the unit ball is compact and is the ball's trace on its affine hullProved

  Oct 2026

• Preimages of points of a finite-dimensional face and of its reflection lie in a common face of no larger dimensionProved

  Oct 2026

• A nonexpansive bijection of unit balls is linear on the diameter through the preimage of an extreme pointProved

  Oct 2026

• Preimages of extreme points under a nonexpansive bijection of unit balls are extremeProved

  Oct 2026

• A nonexpansive bijection of unit balls is odd at preimages of extreme pointsProved

  Oct 2026

• A nonexpansive surjection of unit balls that maps the sphere into the sphere preserves normsProved

  Oct 2026

• Preimages of unit vectors under a nonexpansive surjection of unit balls are unit vectorsProved

  Oct 2026

• A nonexpansive map of a unit ball onto a unit ball does not increase normsProved

  Oct 2026

• A nonexpansive map of a unit ball onto a unit ball fixes the originProved

  Oct 2026

• A point of a convex set outside its relative interior lies in a face of smaller dimensionProved

  Oct 2026

• No retraction of a closed ball onto its boundary sphereProved

  Oct 2026

• Invariance of dimensionProved

  Oct 2026

• Quantitative common-face rigidity: $\|H-I\|_\infty\le\min\{2,4\varepsilon^{1/3}\}$Proved

  Oct 2026

• The Krasnoselskii–Mann residual is bounded by a binomial average of ballot probabilitiesProved

  Oct 2026

• Rate of asymptotic regularity for the Krasnoselskii–Mann iteration with constant step (Baillon–Bruck)Proved

  Oct 2026

• A binomial average of ballot probabilities is at most √(2/(π n u))Proved

  Oct 2026

• The unit ball of $c_{00}(\Gamma)$ is plasticProved

  Oct 2026

• Brouwer fixed-point theorem for the cube $[-1,1]^\iota$Proved

  Oct 2026

• The unit ball of $c_0(\Gamma)$ is plasticProved

  Oct 2026

• No collapse at matched pointsProved

  Oct 2026

• Rigidity: an inward nonexpansive bijection whose finitely supported images come from matched points is the identityProved

  Oct 2026

• Invariance of domain in $\mathbb R^\iota$Proved

  Oct 2026

• Nonexpansive bijections between balls of spaces between $c_{00}$ and $c_0$ are signed coordinate bijectionsProved

  Oct 2026

• Cubical Sperner lemma: a fully labelled small grid simplex existsProved

  Oct 2026

• A codimension-one grid simplex lies in one or two full simplicesProved

  Oct 2026

• An interior codimension-one grid simplex lies in exactly two full simplicesProved

  Oct 2026

Posted 50

• Nonexpansive bijections from balls between c₀₀ and c₀ onto balls between c₀₀ and ℓ∞ are signed permutations, wherever the target matches signsProved

  Oct 2026

• A ball between c₀₀ and ℓ∞ that is the nonexpansive bijective image of a ball between c₀₀ and c₀ lies in c₀Proved

  Oct 2026

• A nonexpansive bijection from a ball between c₀₀ and c₀ onto a ball between c₀₀ and ℓ∞ moves each coordinate inward along a signed permutationProved

  Oct 2026

• A nonexpansive map of a closed bounded set that is isometric on its relative boundary is isometricProved

  Oct 2026

• Two points of a bounded set at equal distances from every point of its relative boundary coincideProved

  Oct 2026

• A nonexpansive ball map that keeps the sphere is isometric between points on whose rays it is homogeneousProved

  Oct 2026

• Normed spaces: a ball whose sphere is covered by finite-dimensional faces is plasticProved

  Oct 2026

• Sphere covered by finite-dimensional faces, normed spaces: every nonexpansive bijection onto the ball is linearProved

  Oct 2026

• A nonexpansive ball bijection that keeps the sphere is homogeneous at every preimage of a point of a finite-dimensional faceProved

  Oct 2026

• If the boundary preimage of a face is compact, extreme and symmetric, the preimage is a face of the same dimensionProved

  Oct 2026

• A compact preimage face is carried onto the face at every scaleProved

  Oct 2026

• A finite-dimensional face of the unit ball is compact and is the ball's trace on its affine hullProved

  Oct 2026

• The reflected preimage of a face is the cone over the boundary preimage from any interior preimageProved

  Oct 2026

• If the smaller faces behave, the preimage of the relative boundary of a face is relatively compactProved

  Oct 2026

• A face whose boundary preimage is relatively compact is carried onto a face of the same dimensionProved

  Oct 2026

• Preimages of points of a finite-dimensional face and of its reflection lie in a common face of no larger dimensionProved

  Oct 2026

• A one-point face of the target ball is carried onto a one-point faceProved

  Oct 2026

• A point of a set lies in its relative interior iff a small ball of its direction space around the point stays in the setProved

  Oct 2026

• A nonempty compact convex set with finite-dimensional span is a closed ball in its direction spaceProved

  Oct 2026

• A continuous self-map of a closed ball fixing its boundary sphere is onto the ballProved

  Oct 2026

• A convex set mapped continuously and injectively into a finite-dimensional affine subspace has no larger dimensionProved

  Oct 2026

• Continuous injections between convex sets of equal finite dimension preserve relative interiorsProved

  Oct 2026

• Invariance of domain between finite-dimensional normed spaces of equal dimensionProved

  Oct 2026

• A ball whose sphere is covered by finite-dimensional faces is plasticProved

  Oct 2026

• Sphere covered by finite-dimensional faces: every nonexpansive bijection onto the ball is linearProved

  Oct 2026

• Every finite-dimensional face of the target ball is carried onto a face of the same dimensionProved

  Oct 2026

• Faces of the unit ball contained in the sphere, and the conclusions of the face theoremDefinition

  Oct 2026

• A nonexpansive surjection of unit balls that maps the sphere into the sphere preserves normsProved

  Oct 2026

• A nonexpansive bijection of unit balls is linear on the diameter through the preimage of an extreme pointProved

  Oct 2026

• Preimages of extreme points under a nonexpansive bijection of unit balls are extremeProved

  Oct 2026

• A nonexpansive bijection of unit balls is odd at preimages of extreme pointsProved

  Oct 2026

• Preimages of unit vectors under a nonexpansive surjection of unit balls are unit vectorsProved

  Oct 2026

• A nonexpansive map of a unit ball onto a unit ball does not increase normsProved

  Oct 2026

• A nonexpansive map of a unit ball onto a unit ball fixes the originProved

  Oct 2026

• No retraction of a closed ball onto its boundary sphereProved

  Oct 2026

• A point of a convex set outside its relative interior lies in a face of smaller dimensionProved

  Oct 2026

• Invariance of dimensionProved

  Oct 2026

• Rate of asymptotic regularity for the Krasnoselskii–Mann iteration with constant step (Baillon–Bruck)Proved

  Oct 2026

• A binomial average of ballot probabilities is at most √(2/(π n u))Proved

  Oct 2026

• The Krasnoselskii–Mann residual is bounded by a binomial average of ballot probabilitiesProved

  Oct 2026

• An interior codimension-one grid simplex lies in exactly two full simplicesProved

  Oct 2026

• Facets of a full grid simplex are its vertex deletionsProved

  Oct 2026

• A codimension-one grid simplex lies in one or two full simplicesProved

  Oct 2026

• Cubical Sperner lemma: a fully labelled small grid simplex existsProved

  Oct 2026

• Cubical Sperner: labelled grids, grid simplices and their facesDefinition

  Oct 2026

• Normal form: after a signed coordinate bijection every coordinate moves inwardProved

  Oct 2026

• No collapse at matched pointsProved

  Oct 2026

• Rigidity: an inward nonexpansive bijection whose finitely supported images come from matched points is the identityProved

  Oct 2026

• Brouwer fixed-point theorem for the cube $[-1,1]^\iota$Proved

  Oct 2026

• Invariance of domain in $\mathbb R^\iota$Proved

  Oct 2026