Geometrical interpretation of conditions

Question

Let $w = \displaystyle{\sum_{i=1}^n \beta_i x_i }$, with $x_1, \dots , x_n \in \bf R^p$$.

Let $y_i \in \{ -1, 1\}$.

I would like to know what is the geometrical interpretations of the three followings conditions :

• $y_i \langle w, x_i \rangle > 1 \implies \beta_i = 0$
• $y_i \langle w, x_i \rangle < 1 \implies \beta_i = y_iC$
• $y_i \langle w, x_i \rangle = 1 \implies 0 \le \beta_i y_i \le C$

for some constant $C$.

Any help would be apreciated.