Dielectric Potential of Air vs Vacuum and Atmospheric Pressure: Relationship Explained

What is the relationship between dielectric potential of air between vacuum and atmospheric pressure?

I think you are asking about the dielectric breakdown voltage of air versus pressure. Pachen's Law describes the breakdown voltage characteristics of gases in general. As you reduce pressure, the breakdown voltage decreases until you reach a point, called the Paschen Minimum. Further decreases in pressure cause the breakdown voltage to markedly increase, such that at very low pressures (high vacuum), the breakdown strength becomes very high. See:
"Your text to link here...(Pachen's Law)":http://en.wikipedia.org/wiki/Paschen's_law
"Your text to link here...(Breakdown Voltage in air vs pressure (1" gap))":http://www.highvoltageconnection.com/articles/paschen-curve.html

if you mean dielectric constant, then for a vacuum it's 1 and for air it's slightly more than 1 (around 1.00059).