Al-Multazim is the part of Ka'ba between its door and the Black Stone. After drinking the water of Zamzam, it is recommended that one should make special supplications at Al-Multazim.
Ibn 'Abbas reported that he used to stand between theBlack Stone corner and the Ka'ba door and said, "The space between the Black Stone and the door of the Ka'bah is called 'AlMultazim' and whoever stands here and supplicates to Allah, he gives him whatever he supplicates for." (Al-Baihaqi)
'Amr bin Shu'aib reported from his father, and he from his father that he said, "I have seen the Prophet (peace be upon him) placing his face and chest against Al-Multazim."
(Some scholars including Bukhari say Al-Hateem is Al-Multazim. To support their view they cite the hadith of the Night Journey in which the Prophet (peace be upon him) said "While I was asleep in (the area called) ' al-Hateem" or also called "al-Hijr...")