Managing Seats at a Communal Table

Description | Communal tables for restaurants are a big emerging trend in quick-service restaurants, and cafeterias. These are large tables shared by people sitting close to others they typically don’t know. Sitting close to someone unknown may be fun, but sitting too close may be embarrassing. This leads to our management problem:

Suppose N >= 2 people are entering in the restaurant, one after the other, and they are being sit at a communal table. People are so thin that they can be represented as points.

The table may be
a1) circular, or
a2) rectangular, with dimensions ax1.

The number N may be
b1) fixed, and then we seek the locations of the N diners maximizing the minimal Euclidean distance among them, or
b2) random, with known probability distribution (for example, P(N=n) = ½^(n-1), n=2,3, ...), and then we seek the locations of the diners maximizing the expectation of the minimal Euclidean distance among them. Can you help the restaurant?

Mathematical background | Optimization.