Plato would wholeheartedly disagree with the formulation Beauty is in the eye of the beholder. With almost all of his philosophy, you need to have a solid grasp on the Theory of Forms.
Plato defines beauty as the object of someone’s love (eros). Love in essence is attention to something, which creates a deep connection between the lover and loved. Beauty is ultimately what causes someone to love a thing. Plato did not mean false feelings, but a deep objective truth. Beauty is what attracts, like a sweet-smelling flower. Love then is the moving agent towards that beauty.
In the Symposium, Plato describes the ascent to Beauty (which is similar to the ascent to truth).
Climbing the ladder of Beauty
First we come to love lower things, bodies, then to particular minds, then to minds in general, then to laws and practices, to knowledge, and then ultimately, the Form of Beauty. Each step coming closer to the ultimate form.
But what is this form of beauty? Is it in the beautiful thing? No, Plato would say that this Form (of Beauty itself) is what gives beauty to beautiful things. The beautiful thing does not have beauty, but receives it from the external form Beauty.
The beauty is objective. I compare this to modern architecture and classical architecture. Because there appears more Beauty (and more Truth) in older architecture, we let it stand longer. Something closer to true beauty has an eternal quality. Modern architecture gets swapped out faster, because it has less truth. It is a fire, that quickly goes out, no fuel to burn it.
The split of subject (observer) and object (observed) is a modern one. There is the objective truth, and the perceiver. Perception is an interplay between both subject and object. Some things cannot be observed by certain people. Ultimately perception of something is an intimate connection between subject and object (the original meaning of the word Intellect).
Objects which have more beauty are more connected to the ultimate source. They are maintained longer. If you are more blind, you see less beauty.
Many ancient philosophers write about the connection between perception of Beauty (and ultimately truth) and how well trained you are. If you cannot see beauty, it says something about the state of your soul.
Beauty in general could be defined as approaching oneness, or harmony. Beautiful things send a unity, a greater whole that the sum of parts.
The philosopher Roger Scruton defined an element of beauty as having the ability to never fill you up. If something is beautiful, like a painting, you wish to return to look at the painting again and linger. It’s not a complete definition, but helps me ground the concept of Beauty.
Much like old architecture, I spend far longer looking at it than the modern varieties of buildings.