I think you are confusing two things here. A linear program has a convex feasible region. The term "convex space" refers to an entire vector space, not to a particular region (such as an LP feasible region) within the space. It basically imposes a restriction on the nature of the norm used in defining the space.

Update:

Is there any simple example, in the context of LP and MIP, to illustrate that?

The term "strictly convex" only applies to functions, not to sets.

Also, for clarifying The distinction between the convex space and the subset of the convex space, the following example would be useful:

Consider the triangle $T$ defined by $x \geq 0$, $y \geq 0$, $x + y > \leq 1$, which is convex. The subset consisting of the vertices, $\{(0,0), (0,1), (1,0)\}$ is not convex. Neither is the set $\{(x,y) > \in T : x^2 + y^2 \geq 1/2\}$.

I think you are confusing two things here. A linear program has a convex feasible region. The term "convex space" refers to an entire vector space, not to a particular region (such as an LP feasible region) within the space. It basically imposes a restriction on the nature of the norm used in defining the space.

Update:

Can we refer to the feasible set of linear programming as strictly convex?

The term "strictly convex" only applies to functions, not to sets.

Can we say any subset of a convex set/space would be a convex set/space?

No. Consider the triangle $T$ defined by $x \geq 0$, $y \geq 0$, $x + y \leq 1$, which is convex. The subset consisting of the vertices, $\{(0,0), (0,1), (1,0)\}$ is not convex. Neither is the set $\{(x,y) \in T : x^2 + y^2 \geq 1/2\}$.

I think you are confusing two things here. A linear program has a convex feasible region. The term "convex space" refers to an entire vector space, not to a particular region (such as an LP feasible region) within the space. It basically imposes a restriction on the nature of the norm used in defining the space.

I think you are confusing two things here. A linear program has a convex feasible region. The term "convex space" refers to an entire vector space, not to a particular region (such as an LP feasible region) within the space. It basically imposes a restriction on the nature of the norm used in defining the space.

Update:

Is there any simple example, in the context of LP and MIP, to illustrate that?

The term "strictly convex" only applies to functions, not to sets.

Also, for clarifying The distinction between the convex space and the subset of the convex space, the following example would be useful:

Consider the triangle $T$ defined by $x \geq 0$, $y \geq 0$, $x + y > \leq 1$, which is convex. The subset consisting of the vertices, $\{(0,0), (0,1), (1,0)\}$ is not convex. Neither is the set $\{(x,y) > \in T : x^2 + y^2 \geq 1/2\}$.