Adam Paszkiewicz: On convex Peano curve, connected methods and problem

HyperComplex Seminar, 17 Sep. 2024
Abstract. We present the following main result. For any closed ball T in Cartesian space RD there exists a continuous surjection f : [0, 1] → T, such that the images f ([α, β]) is convex and non-empty interior for any non-degenerate interval [α, β] ⊂ [0, 1].
The construction of such function f consists in some investigation of covers (t(k); 1 ⩽ k ⩽ m) of the ball T convex in the following sens. The unions t(k)∪. . .∪t(l) are convex for any indices k, l ∈ {1, . . . , m}, k ⩽ l. Developed new methods of convex analysis can be used in many similar problems for important classes of functions. Some conjectures will be discussed more precisely.
Results are new, can be found in [1] and [2].
Keywords and phrases: Peano curve, an transforms, convex sets
References
[1] A. Paszkiewicz, The convex Peano curve does exists, ArXiv: 247.0316
[2] D. Prusinowski, Wypukla krzywa Peano wypelniaj¡ca kulę w RD , doctoral dissertation, in preparation.