On Generalized “ham Sandwich” Theorems

In this short note we utilize the Borsuk-Ulam Anitpodal Theorem to present a simple proof of the following generalization of the “Ham Sandwich Theorem”: Let A1, . . . , Am ⊆ R be subsets with finite Lebesgue measure. Then, for any sequence f0, . . . , fm of R-linearly independent polynomials in the polynomial ring R[X1, . . . , Xn] there are real numbers λ0, . . . , λm, not all zero, such that the real affine variety {x ∈ R; λ0f0(x)+ · · ·+λmfm(x) = 0} simultaneously bisects each of subsets Ak, k = 1, . . . ,m. Then some its applications are studied. The Borsuk-Ulam Antipodal Theorem (see e.g. [2, 12]) is the first really striking fact discovered in topology after the initial contributions of Poincaré and its fundamental role shows an enormous influence on mathematical research. A deep theory evolved from this result, including a large number of applications and a broad variety of diverse generalizations. In particular, as it was shown in [9], an interrelation between topology and geometry can be established by means of an appropriate version of the famous “Ham Sandwich” Theorem deduced from the Borsuk-Ulam Antipodal Theorem. It was pointed out in [6] that an existence of common hyperplane medians for random vectors can be proved from the “Ham Sandwich” Theorem as well. The presented main result is probably known to some experts but its proof is much simpler than others in the literature and some consequences are easily deduced. Our paper grew up to answer the question posed in [6]; that is of which curves or manifolds other than straight lines or hyperplanes can serve as common medians for random vectors. To settle that question we make use of the result which is presented in later given Theorem 4. Let R be the field of real numbers, R the n-Euclidean space and S the nsphere. The following theorem is well known (see e.g. [3, p.79] or [4, p.287]). 2000 Mathematics Subject Classification. Primary 58C07; Secondary 12D10, 14P05.