The image-based claim didn't link to any articles or other corroborating information, saying only:
Massaging your breasts with olive oil is an excellent technique to firm sagging breasts. Olive oil is a rich source of antioxidants and fatty acids that can reverse the damage caused by free radicals and prevent sagging breasts. Plus, it will help improve the skin tone and texture.
According to the claim, topical application of olive oil either circumvents or reverses loss of elasticity in breasts. No specific reason was provided for the claim that olive oil, versus any other type of oil, could replicate the effect shown.
The efficacy of topical treatments for sagging breasts or loss of skin elasticity is often discussed in cosmetic circles, thanks in part to the cost and invasiveness of surgical approaches. But research demonstrating any single topical ingredient can prevent (much less reverse) loss of skin elasticity is notably skimpy:
Antioxidant ointments, creams, and lotions may help reduce the risk of wrinkles and protect against sun damage. Unlike sunscreens, they build up in the skin and are not washed away, so the protection may last longer. Selenium, coenzyme Q10 (CoQ10), and alpha-lipoic acid are antioxidants that can be applied to the skin. Evidence of their benefit is limited and more studies in humans are needed. More commonly used antioxidants are described below. Your dermatologist (skin doctor) can tell you which product is right for you.
Research from 2012 indicates that anxioxidants could potentially prevent damage to facial skin, but evidence was "limited" and reversal was not referenced. Only 1.5 percent of olive oil contains antioxidants (and that fraction is not composed entirely of antioxidants). Other foods are described as being higher in antioxidants, but no memes recommended rubbing a berry paste onto breasts for firmness. Research overwhelmingly examined efficacy of topical treatment for facial skin, which lacks the weight and density of breast tissue. And although topicals designed for facial skin need only penetrate a few dermal layers, any product designed or intended for use on breasts would have to enter the breast tissue to "lift" or prevent elasticity loss (a far taller order).
A blog post written by Miami-based cosmetic surgeon Dr. Ary Krau directly addresses the question of whether any specific topical substance (including olive oil) had ever shown promise to improve loss of elasticity for breast tissue in clinical research:
Unfortunately, no cream or lotion has been clinically proven or FDA-approved to lift the breasts into a higher position. Don’t waste your money on any product that claims to improve your breast position.
While genetics play a big role in breast sagging, there are a few things you can do to prevent it in the first place. Try not to fluctuate greatly in body weight; gaining and losing a lot of weight in a short period of time can take its toll on the breasts. Be diligent about wearing sunscreen on your chest and décolletage area to protect the skin from the sun’s rays. Wear a supportive bra as much as possible, especially when performing high-intensity aerobic exercise.
We contacted Dr. Krau's office, and a representative for the clinic told us that no effective topical treatment appears to exist. A thorough search of related studies turned up no strong evidence facial skin or breast tissue could be firmed by topical treatments at all. A 5 percent concentration of Vitamin C (not olive oil) showed moderate success in treatment of photo-aging in a small 2003 study, but those findings were not relevant to breast tissue or olive oil.
There is research [PDF] (conducted by individuals who contracted with cosmetic medical firms) that indicates some success in the ongoing development of a topical version of botulinum toxin type A (popularly known by the brand name Botox) and intended solely for the face. However, that research only concerns a clinically prescribed substance (botulinum toxin, not a pantry product) and included no indications that it could be used on breast tissue.