  To find the essential nature of quantum theory has been an important problem
for not only theoretical interest but also applications to quantum
technologies. In those studies on quantum foundations, the notion of
uncertainty plays a primary role among several stunning features of quantum
theory. The purpose of this thesis is to investigate fundamental aspects of
uncertainty. In particular, we address this problem focusing on convexity,
which has an operational origin. We first try to reveal why in quantum theory
similar bounds are often obtained for two types of uncertainty relations,
namely, preparation and measurement uncertainty relations. To do this, we
consider uncertainty relations in the most general framework of physics called
generalized probabilistic theories (GPTs). It is proven that some geometric
structures of states connect those two types of uncertainty relations in GPTs
in terms of several expressions such as entropic one. Our result implies what
is essential for the close relation between those uncertainty relations. Then
we consider a broader expression of uncertainty in quantum theory called
quantum incompatibility. Motivated by an operational intuition, we propose and
investigate new quantifications of incompatibility which are related directly
to the convexity of states. It is also shown that there can be observed a
notable phenomenon for those quantities even in the simplest incompatibility
for a pair of mutually unbiased qubit observables. Finally, we study
thermodynamical entropy of mixing in quantum theory, which also can be seen as
a quantification of uncertainty. We consider its operationally natural
extension to GPTs, and then try to characterize how specific the entropy in
quantum theory is. It is shown that the operationally natural entropy is
allowed to exist only in classical and quantum-like theories among a class of
GPTs called regular polygon theories.
