equal
deleted
inserted
replaced
276 qed |
276 qed |
277 qed |
277 qed |
278 qed |
278 qed |
279 qed |
279 qed |
280 |
280 |
281 proposition (*FIXME needs name *) |
281 proposition |
282 fixes S :: "'a::euclidean_space set" |
282 fixes S :: "'a::euclidean_space set" |
283 assumes "compact S" and 0: "0 \<in> rel_interior S" |
283 assumes "compact S" and 0: "0 \<in> rel_interior S" |
284 and star: "\<And>x. x \<in> S \<Longrightarrow> open_segment 0 x \<subseteq> rel_interior S" |
284 and star: "\<And>x. x \<in> S \<Longrightarrow> open_segment 0 x \<subseteq> rel_interior S" |
285 shows starlike_compact_projective1_0: |
285 shows starlike_compact_projective1_0: |
286 "S - rel_interior S homeomorphic sphere 0 1 \<inter> affine hull S" |
286 "S - rel_interior S homeomorphic sphere 0 1 \<inter> affine hull S" |
538 apply (subst homeomorphic_sym) |
538 apply (subst homeomorphic_sym) |
539 apply (rule homeomorphic_compact [OF co01 cont_nosp2 [unfolded o_def] im_cball inj_cball]) |
539 apply (rule homeomorphic_compact [OF co01 cont_nosp2 [unfolded o_def] im_cball inj_cball]) |
540 done |
540 done |
541 qed |
541 qed |
542 |
542 |
543 corollary (* FIXME needs name *) |
543 corollary |
544 fixes S :: "'a::euclidean_space set" |
544 fixes S :: "'a::euclidean_space set" |
545 assumes "compact S" and a: "a \<in> rel_interior S" |
545 assumes "compact S" and a: "a \<in> rel_interior S" |
546 and star: "\<And>x. x \<in> S \<Longrightarrow> open_segment a x \<subseteq> rel_interior S" |
546 and star: "\<And>x. x \<in> S \<Longrightarrow> open_segment a x \<subseteq> rel_interior S" |
547 shows starlike_compact_projective1: |
547 shows starlike_compact_projective1: |
548 "S - rel_interior S homeomorphic sphere a 1 \<inter> affine hull S" |
548 "S - rel_interior S homeomorphic sphere a 1 \<inter> affine hull S" |