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When I issue the command lightning-cli listpeers $peerid i get a feerate perkb of 1012 atm. What feerate is this exactly? Is this the feerate which will be proposed by the other peer for a mutual close? I don't think this is the case.

feerate: An object containing the latest feerate as both perkw and perkb.

I know that i can get the feerates of my node with lcli feerates perkb. With these results atm:

{
   "perkb": {
      "opening": 1012,
      "mutual_close": 1012,
      "unilateral_close": 1012,
      "delayed_to_us": 1012,
      "htlc_resolution": 1012,
      "penalty": 1012,
      "min_acceptable": 1012,
      "max_acceptable": 88300
   },
   "onchain_fee_estimates": {
      "opening_channel_satoshis": 177,
      "mutual_close_satoshis": 170,
      "unilateral_close_satoshis": 151,
      "htlc_timeout_satoshis": 168,
      "htlc_success_satoshis": 178
   }
}

That means my node will propose 1012 as a starting fee for a mutual close. But what is the starting fee of the other node?

0

You won't know what the other node proposes unless some one starts the process of closing the channel and then your node will see what they propose and start digging into the estimation.

In general one can access the man-page with lightning-cli help feerates (or in your case with lcli help feerates) from there we find further information with respect to negotiation of fees.

It says that the closing negotiation start with your node proposing mutual_close fess as you suggested. I quote:

mutual_close - the starting feerate used in mutual close negotiation.
Note that since mutual close is a negotiation, the actual feerate used in 
mutual close will be somewhere between this and the corresponding mutual
close feerate of the peer.

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