Update:
Linear programming problems (LP) have a convex space, precisely vector space, such as a convex feasible region as pointed out by @prubin. Also, they may have either unique or multiple solutions. (there are already other types).
On the other hand, there is still a concept named strictly convex that interprets:
a strictly convex space is a normed vector space (X, || ||) for which the closed unit ball is a strictly convex set. Put another way, a strictly convex space is one for which, given any two distinct points x and y on the unit sphere ∂B (i.e. the boundary of the unit ball B of X), the segment joining x and y meets ∂B only at x and y. Strict convexity is somewhere between an inner product space (all inner product spaces being strictly convex) and a general normed space in terms of structure. It also guarantees the uniqueness of the best approximation to an element in X (strictly convex) out of a convex subspace Y, provided that such an approximation exists.
The distinction between these two concepts comes in the following (for more details please, see this link):
A function $f: \mathbb{R} \to \mathbb{R}$ is convex if for all $x,y > \in \mathbb{R}$ and for all $\lambda \in (0,1)$ the following holds: $$f(\lambda x +(1-\lambda)y) \leq \lambda f(x) +(1-\lambda) f(y)$$
Geometrically this means that the line through two points $f(x)$ and $f(y)$ on the graph is always above the graph between $x$ and $y$.
We say that $f$ is strictly convex if the above inequality holds strictly, i.e. $$f(\lambda x +(1-\lambda)y) < \lambda f(x) > +(1-\lambda) f(y)$$
Since I would like to know, 1) What does exactly it mean? 2) Is there any simple example, in the context of LP and MIP, to illustrate that? 3)If an LP has a such space, might it have only a unique solution?
