Zdravím. Jaký je v tom rozdíl ? On the edge of je především pro něco negativního? Děkuji

The difference between on the edge / verge / (a aby toho nebylo málo) point / brink / cusp (of something, of doing something) isn't about positivity or negativity, it's primarily about the physical or metaphorical sense in the context.

• On the edge/verge/po­int/cusp/brink of the pavement sat a dog.

In the metaphorical sense, verge is usually preferred and usually sounds more idiomatic.

• I was on the verge/point/brin­k/cusp /edge of calling an ambulance when the pain suddenly stopped and I felt fine again.

I recommend that you spend a few minutes on Google to see how they're used in different contexts – and look for “real” examples from native UK or US sources, not just dictionary entries.

I would agree that “on the edge” can be used in literal, non-metaphorical contexts whereas “on the verge” would not work:

• “Be careful. The glass is near the edge of the counter. It might fall.”
• NOT: “on the verge of the counter” = nesmysl

“On the edge” can also be idiomatic:

• I was on the edge of my seat = it was really exciting/suspen­seful
• She was on the edge of despair

Verge doesn't have to be used in negative or dramatic contexts, but it often is:

• I was on the verge of a nervous breakdown.
• He was on the verge of bankruptcy.
• I was on the verge of divorcing my husband, but then…
• She was on the verge of tears when…

Dramatic yet positive:

• We were on the verge of a scientific breakthrough.