Adversarial risk in high dimension


I have a question regarding adversarial risk in high dimension. Based on the first example provided in lecture 9c my takeaway is that there is a tradeoff between the standard and adversarial risks, which is also the case in the second example provided regarding the spherical cap i.e.

$$R_{\epsilon}(f) = 1 - p$$

What I do not understand is, why would an increase in p (i.e. the standard risk), which results in a decrease in the boundary \(\alpha\) so it is closer to the equator, reduce the adversarial risk? I am not sure what I am misunderstanding but any clarification would be much appreciated!

Thanks in advance for your time and apologies for the strange formatting (I tried using the latex equation block button but it does not seem to work)

[admin: added block \(\LaTeX\) as requested. Button is broken indeed, will fix asap. Manual formatting guide here. Bugs/improvements welcome here.]

I was also struggling with latex. There are two ways to write down equations, the double $ works only when used in a new line. To use in line you need to surround the equation with brackets but escape both brackets first.
Example: \ ( f ( x ) \ )

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