A utility between 0 and 1 effectively plays the same role as “negative utilities” do under regular arithmetic expected value. So a geometric perspective can work if you map what you would ordinarily consider to be negative utilities to the interval 0 to 1, and restrict utility to .
But why would you? It doesn’t offer any benefit over regular artihmetic expected value calculations.
I’m surprised you didn’t bring up the most commonly cited defintions of capitalism and socialism.
Capitalism is the private ownership of the means of production.
Socialism is the public ownership of the means of production.
Whilst there’s a great deal of variation in how people envision their form of captialism/ socialism, the above are the generally agreed upon dictionary definitions of the two economic systems.