Peng Zhou

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====== 2026-03-27 ======

In the nicest setting, max of smooth psh function is still psh, but with kink when the dominant term switch over. To solve this problem, people developed softmax, which is a smearing of max function. When we softmax a bunch of psh function, the outcome is smooth and psh. 

Another application is the following: suppose we have a bunch of locally defined psh function $u_\alpha$, living on some locally finite open cover $\Omega_\alpha$ (say extending continuously to the closure of $\Omega_\alpha$). If we take max of these whole collection of functions, that certainly does not make sense. If we take max at each point $z$, then the problem is that if $z$ moves out of the boundary of certain $\Omega_\beta$, $u_\beta$ will suddenly not avaiable for doing max, it would be a disaster when the 'weight bearer' of the group suddenly leave, we would have a cliff fall over. The only case where everything is safe, is when $u_\beta$ is already relatively 'retired' near the boundary $\Omega_\beta$, as the real work is taken up by some other $u_\alpha$, then there is no problem. Then, you can take pointwise max, the thing will still be continuous psh. 

Now, we don't want to do convolution to regularize continuous psh. We want to do softmax. The problem with softmax is that, each term needs a room of epsilon to smooth over. This is usually no problem, the fuzzy uncertainty for $u_\beta$ by $\eta_\beta$ is tolerable, if the bump-up of $u_\beta$ still won't catch the low-day of the best $u_\alpha$, then it is safe to retire, byebye safe trip. 

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Next, we consider Richberg's theorem. Input a strictly psh function on a manifold $X$
  * Do a locally finite covering of $\Omega$, so that each $\Omega_\alpha$ is in a coordinate patch
  * Do some convolution smoothing for each patch $u_\alpha$. 
  *