The Early History of the Ham Sandwich Theorem

Beyer, W. A.; Żardecki, A.

doi:10.1080/00029890.2004.11920050

Cited by 36 publications

(22 citation statements)

References 2 publications

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“…As noted above, Theorem 2.1 recovers the Ham Sandwich Theorem when m = n, and in fact our proof of Theorem 2.1 reduces to the classical geometric proof of the Ham Sandwich Theorem as attributed to Banach by Steinhaus [8] in this case. For m < n, our best estimates occur when Span(RP n−1 ) is largest.…”

Section: Bisections By Orthogonal Hyperplanessupporting

confidence: 56%

Mass partitions via equivariant sections of Stiefel bundles

Simon¹

2018

Filomat

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We consider a geometric combinatorial problem naturally associated to the geometric topology of certain spherical space forms. Given a collection of m mass distributions on R n , the existence of k affinely independent regular q-fans, each of which equipartitions each of the measures, can in many cases be deduced from the existence of a Zq-equivariant section of the Stiefel bundle V k (F n ) over S(F n ), where V k (F n ) is the Stiefel manifold of all orthonormal k-frames in F n , F = R or C, and S(F n ) is the corresponding unit sphere. For example, the parallelizability of RP n when n = 2, 4, or 8 implies that any two masses on R n can be simultaneously bisected by each of (n − 1) pairwise-orthogonal hyperplanes, while when q = 3 or 4, the triviality of the circle bundle V2(C 2 )/Zq over the standard Lens Spaces L 3 (q) yields that for any mass on R 4 , there exist a pair of complex orthogonal regular q-fans, each of which equipartitions the mass.

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Section: Bisections By Orthogonal Hyperplanessupporting

confidence: 56%

Mass partitions via equivariant sections of Stiefel bundles

Simon¹

2018

Filomat

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“…It is noteworthy that BUT and its variants talk about projections, connections, not about cause-and-effect relationships . Briefly, BUT states that, if a single point on a circumference projects to a higher spatial dimension, it gives rise to two antipodal points with matching description on a sphere, and vice versa ( Figure 1A) (Borsuk, 1933;Marsaglia, 1972;Matoušek, 2003;Beyer, 2004). This means that the two antipodal points are assessed at one level of observation in terms of description, while a single point is assessed at a lower level (Tozzi 2016b), i.e., point location vs. point description.…”

Section: Methodsmentioning

confidence: 99%

“…And these mappings described by BUT can sometimes be reversed (inversed) to achieve a form of pullback from descriptions to sources of descriptions, from a simplified view to multiple views of the same object. Indeed, a new form of shape theory (called homotopy) discovered by K. Borsuk makes it possible to assess the properties that are preserved through deformation, stretching and twisting of objects (Beyer, 2004;Manetti, 2015). Homotopy is a theory of shape deformation (Borsuk, 1971;Borsuk and Dydak, 1980), e.g., how some shapes can be deformed into other shapes.…”

Section: Methodsmentioning

confidence: 99%

A Topological/Ecological Approach to Perception

Tozzi

Peters

Ramanna

2016

Preprint

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During the exploration of the surrounding environment, the brain links together external inputs, giving rise to perception of a persisting object. During imaginative processes, the same object can be recalled in mind even if it is out of sight. In this proof of concept study, Borsuk's theory of shape and the Borsuk-Ulam theorem provide a mathematical foundation for Gibson's notion of persistence perception. Gibson's ecological theory of perception accounts for our knowledge of world objects by borrowing a concept of invariance in topology. A series of transformations can be gradually applied to a pattern, in particular to the shape of an object, without affecting its invariant properties, such as boundedness of parts of a visual scene. High-level representations of objects in our environment are mapped to simplified views (our interpretations) of the objects, in order to construct a symbolic representation of the environment. The representations can be projected continuously to an ecological object that we have seen and continue to see, thanks to the mapping from shapes in our memory to shapes in Euclidean space.Theoretical physics teaches us that the intimate micro-structure of the world consists of elements in perpetual movement, interacting with each other in a framework of probabilities, energy fields and vacuum. However, when we see a segment in a visual scene in the environment, we perceive elements seemingly melted together in a single complex of sensations. To make an example, we are able to detect, in the indistinctness of a rural scene at sunrise time, an increasingly distinct world of trees, hills, valleys and moving particles, e.g., birds flying from one tree to another. In effect, we appear to be sewing pieces of a changing scene together. How does the brain join different significant elements, giving rise to a single, stable perception of a scene? How does the brain imagine or recall objects that are out of sight? These two problems need to be tackled not at the microscopic atomic or sub-atomic level, nor at the galactic level, but at an intermediate macroscopic level where living beings stand. A noteworthy approach to understanding the mechanisms of direct perception is to start at the ecological level. Indeed, this article introduces a shape-based explanation of J.J. Gibson's theory of persistence perception (Gibson, 1950). Throughout the second half of the twentieth century, James J. Gibson (1966Gibson ( , 1971Gibson ( , 1979) developed a unique theory in perceptual science, namely the ecological theory of perception (ETP), which stresses the importance of the relationships of the individual and the environment (Heft, 1997). The foundation for perception is ambient, ecologically available, direct information, as opposed to peripheral or internal sensations. The human (and/or animal) individual is embedded in the surrounding environment and the perception is strictly linked to searching movements, because changing awareness of the connectedness of the components of a scene occurs in situ, and not...

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“…The quintessential measure partitioning result is the Ham Sandwich theorem, which was conjectured by Steinhaus and proved subsequently by Banach in 1938 (consult for example ). The Ham Sandwich theorem is one of the most widely known consequences of the Borsuk–Ulam theorem.…”

Section: Introductionmentioning

confidence: 94%

Thieves can make sandwiches

Blagojević

Soberón

2017

Bull. London Math. Soc.

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We prove a common generalization of the Ham Sandwich theorem and Alon's Necklace Splitting theorem. Our main results show the existence of fair distributions of m measures in R d among r thieves using roughly mr/d convex pieces, even in the cases when m is larger than the dimension. The main proof relies on a construction of a geometric realization of the topological join of two spaces of partitions of R d into convex parts, and the computation of the Fadell-Husseini ideal valued index of the resulting spaces.

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The Early History of the Ham Sandwich Theorem

Cited by 36 publications

References 2 publications

Mass partitions via equivariant sections of Stiefel bundles

Mass partitions via equivariant sections of Stiefel bundles

A Topological/Ecological Approach to Perception

Thieves can make sandwiches

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