Problems Related To Isosceles And Pythagorean Orthogonality

Isosceles orthogonality and Pythagorean orthogonality are two orthogonalitieswhich do not have homogeneity and additivity in normed linear spaces, so the setisosceles orthogonal or Pythagorean orthogonal to a given vector often have morecomplex characterization than a hyperplane through origin o. Study thecharacterizations of such sets can help us grasp the geometric properties of the spacespreferably. Athough some scholars have done some work in this direction, somebasic problems have not been solved.This paper study the path-connectivity of the above two kinds of sets and itssubsets.On the one hand, we study the path-connectivity of the intersection of the radialprojections of bisectors and spheres. It is proved that, for an arbitrary unit vector xin a normed linear space whose dimension is not less than3and each non-negativereal number γ not greater than1, the set of vectors isosceles orthogonal to xhaving norm γ is path-connected. Using this result, the path-connectivity of radialprojections of bisectors is proved. It is shown that, for an arbitrary unit vector in a3dimensional normed linear space, there exists a normal base containing this vectorwhose elements are pairwise isosceles orthogonal.On the other hand, we prove the set pythagorean orthogonal to a given vectoris path-connected. Using this result, prove the path-connectivity of the radialprojections of such sets. In addition, we also introduce and study a new constantassociated with such sets.