Riemannian manifolds are KKM spaces

Abstract

Let (M, g) be a complete, finite-dimensional Riemannian manifold. Based on the fact that any geodesic convex subset of M is a KKM space, we establish the KKM theory on such subsets originated from the Knaster-Kuratowski-Mazurkiewicz theorem in 1929.

Minimax in Geodesic Metric Spaces: Sion's Theorem and Algorithms

“…Our analysis generalizes their result by removing the reliance on Riemannian differential structure along with other additional conditions. Our work is closest to Park (2019Park ( , 2010, who shows that Sion's theorem can be established for the novel KKM space that subsumes Hadamard manifolds. Nevertheless, it remains difficult to verify whether a given geometry satisfies the KKM conditions.…”

Elements of Convex Geometry in Hadamard Manifolds with Application to Equilibrium Problems

“…In addition, we also present a proof for a version of the KKM's lemma using the Helly's theorem (see Theorem 3.4). It is worth mentioning that based on this version of KKM's lemma presented in our paper, it is possible to obtain the versions considered in [19,58,49]. However, since this topic, which deals with combinatorial convexity, demonstrates an interdisciplinary character, we understand that the version of the KKM's lemma presented in this paper can be made more useful.…”

“…(ii) From this version of the KKM lemma described above, it is possible to obtain the versions considered in [19,58,49] where, in item (b) above, n + 1 is replaced by a certain variable natural m. However, since this topic dealing with combinatorial convexity shows to have an interdisciplinary character, we understand that this version of the KKM lemma presented in the last result can be made more useful in practice due to the smaller number for checking the condition established in item (b).…”

“…As already highlighted in the introduction of the paper, the KKM lemma was used as a tool to establish result of existence for equilibrium problems; see, for example, [19,49,8,4] for references dealing with this topic in the Riemannian setting, whose approaches extend the results of existence established directly to some important particular instances such as variational inequality [47,44,42] and Nash equilibrium points [37,38]. Limiting the reference [8], which we believe to be the most recent on the topic, it is possible to notice an important connection between combinatorial convexity, established by the KKM lemma, and "variational rationality" approach of human behavior, characterizing the relevance of the theme to the interdisciplinary research.…”

“…Regardless of this in the linear setting, we would like to mention about reference [45], which does not only present a beautiful background on KKM's lemma with a rich list of references on the subject, but also presents important connections with the theorem of Helly and Carathéodory. In the Riemannian context, we highlight the following references that deal the KKM's lemma [19,58,49]. As a special case, see [48,50], in which an approach in other contexts that has Hadamard manifold as a particular case have been presented.…”