# Relative interior requirement in Slater's condition

I'm reading Convex Optimization by Boyd and Vandenberghe. This is how they describe Slater's condition:

What I don't understand is why it is necessary to enforce that $$x$$ be in the relative interior of $$\mathcal{D}$$, where

$$\mathcal{D} := \bigcap_{i = 1}^m \text{dom}(f_i),$$

i.e., it is the intersection of the domains of the $$f_i$$'s. Doesn't the fact $$f_i(x) < 0$$ for all $$i$$ immediately imply that $$x \in \text{relint}(\mathcal{D})$$? So why state the theorem this way?

As a side note, Wikipedia gives the exact same seemingly extraneous condition:

Please let me know what I'm misunderstanding. Why is this additional condition necessary? Thank you!

EDIT: I probably have some fundamental misunderstanding, so I'm adding this edit to be more clear about what I'm thinking. If I'm not wrong, the fact that the constraints $$f_i$$ are convex means that the intersection of the spaces spelled out by $$f_i(x) \le 0$$ for all $$i = 1, \dots, m$$ is convex. The feasible region is then the intersection of this convex space with an affine space, that being the affine space spelled out by $$Ax = b$$.

It seems to me that any point $$x$$ that satisfies $$f_i(x) < 0$$ will not only be in $$\text{relint} \mathcal{D}$$, it will even be in the relative interior of the feasible region described above. This is just because the convex functions $$f_i(x) = 0$$ form the boundary of the feasible region (in a semi-intuitive sense). The fact that $$x$$ satisfies $$f_i(x) < 0$$ for all $$i$$ means that it will be an interior point since it doesn't lie on the boundary.

• How does the sign of f(x*) imply that it is in relint D? The points belonging to relint D are determined by D and the space in which it is embedded, not by the values of f in any way. May 16, 2021 at 8:42
• @AmitHochman I just added a clarifying edit at the end to better show what I'm thinking. Please let me know what I'm misunderstanding. Maybe my intuition that strict inequalities --> interior point is wrong in a general setting? May 16, 2021 at 15:58
• Are you sure that's an extra condition and not just the authors explaining the condition? (Also, Wikipedia most likely just copied from that standard reference.) May 16, 2021 at 21:22
• To be clear: the relint condition is the general condition that holds in for an arbitrary constraint $x \in \mathcal{D}$, even in Banach spaces. For the special case that the constraint consists of finitely many scalar inequalities, (5.26) is indeed sufficient. May 16, 2021 at 21:25
• Also, recall the definition of $\mathcal{D}$, which includes the equality constraints $h_i$ (which is why the relint is important) as well as the objective function $f_0$! May 16, 2021 at 21:27